The problems of the All-Soviet-Union mathematical competitions 1961-1986
This file contains the problems, suggested for solving at the Russian national
mathematical competitions (final part). I've posted this stuff in a number of
articles in rec.puzzles, but I have got many requests for the missing parts.
So, I have decided to put this material here, having provided it with the
answers to the common questions.
collected articles start
I'm going to send some problems from the book
Vasil'ev N.B, Egorov A.A. "The problems of the All-Soviet-Union
mathematical competitions",-Moscow.:Nauka. 1988 ISBN 5-02-013730-8.
(in Russian).
Those problems were submitted for the solving on the competition
between the pupils of 8, 9, or 10 forms for 4 hours. So they do not
contain something of the advanced topics, -- all of them can be solved
by a schoolboy. They can not go out of the common school plan bounds.
Most of the problems are original and the book contains all the
necessary references. I am not going to translate all the book, so I
shall not send the solutions. Please, accept those messages as they
are, to say more exactly -- as I can. I have to do my job, and this is
hobby only, but nevertheless, that should be enjoyable to solve those
problems.
"Nobody can embrace the unembraceable."
Kozma Prutkov. (beginning of the XIX c.)
May be those postings are just a harassment, but I hope, that most
of You will not only enjoy the problems solving, but will be able
to use them in Your work with the students.
"Zeal overcomes everything."
"Sometimes zeal overcomes even the common sense."
Kozma Prutkov.
There are some wonderful books in Russian, that have not been
translated into English yet, for example,
"Problems of the Moscow mathematical competitions",-compiled
by G.A.Galperin, A.K.Tolpygo.,-Moscow, Prosveshchenie, 1986.
A.A.Leman "Collection of Moscow mathematical competitions problems",
-Moscow, Prosveshchenie, 1965.
answers for the common questions
WHAT IS THE AGE OF THE PARTICIPANTS?
The Russian pupils start studying being 6-7 years old, so the
pupils of the 8th form are about 14.
WHO PARTICIPATE?
The competition is held in 3 - 4 stages
1. At school - if there are many volunteers.
2. Subregional - if the region is big enough.
3. Regional (in some regions, as in Moscow, Leningrad=Sankt-Petersburg,
Sverdlovsk=Yekaterinburg or Novosibirsk they are even more
interesting and more difficult)
4. Final part, considered in the report.
WHAT ARE THE BEST AND AVERAGE RESULTS?
The winners (2-5) usually give the perfect solution of all the problems
with some shortages.
My personal experience refers to that times, when there were two days
of the final competition. Than the winners solved all the problems of
two days except one problem.
It is very difficult to speak about the average level, because it
depends very much on a region, but most of the participants of that
time solved at least one problem. The problem is not only the
difficulties in the problems themselves, but also in the shortage of
time. They successfully solved the problems before the official
explanation two days later.
MAY I USE THOSE PROBLEMS IN THE SCHOOL PROJECTS?
You don't need MY permission for using those problems. As concerns the
copyright, the usage of all the information in the non-commercial
purposes was never restricted in Russia if it is not related to the
state security. Moreover, the spirit of the competition encourages
everybody to distribute those problems in order to enhance the
mathematical culture of the pupils.
And You only are able to decide whether Your students can solve them.
ARE THERE THE TRANSLATED SOLUTIONS OF THOSE PROBLEMS ELSEWHERE?
Sorry, there exists no complete translation of the cited book.
Besides, the solutions (in comparison with the problems themselves)
belong to the authors, and the translation without their explicit
permission would be their copyright violation.
Given a figure, containing 16 segments.
You should prove that there is no curve, that intersect each segment
exactly once. The curve may be not closed, may intersect itself, but it
is not allowed to touch the segments or to pass through the vertices.
002.
Given a rectangle A1A2A3A4.
Four circles with Ai as their centres
have their radiuses r1, r2, r3, r4;
and r1+r3=r2+r4<d,
where d is a diagonal of the rectangle. Two pairs of the outer common
tangents to {the first and the third} and {the second and the fourth}
circumferences make a quadrangle.
Prove that You can inscribe a circle into that quadrangle.
003.
Prove that among 39 sequential natural numbers there always is a
number with the sum of its digits divisible by 11.
004.
Given a table 4x4.
a)
Find, how 7 stars can be put in its fields in such a way, that
erasing of two arbitrary lines and two columns will always leave at
list one of the stars.
b)
Prove that if there are less than 7 stars, You can always find two
columns and two rows, such, that if You erase them, no star will remain in
the table.
005.
a)
Given a quartet of positive numbers. (a,b,c,d). It is transformed to
the new one according to the rule:
a'=ab; b' =bc; c'=cd; d'=da.
The second one is transformed to the third according to the same rule
and so on.
Prove that if at least one initial number does not
equal 1, than You can never obtain the initial set.
b)
Given a set of 2k (k-th power of two) numbers, equal either to 1 or
to -1. It is transformed as that was in the a) problem (each one is
multiplied by the next, and the last -- by the first.
Prove that You will always finally obtain the set of positive units.
006.
a)
Points A and B move uniformly and with equal angle speed along the
circumferences with Oa and Ob centres
(both clockwise).
Prove that a vertex C of the equilateral triangle ABC also moves
along a certain circumference uniformly.
b)
The distance from the point P to the vertices of the equilateral
triangle ABC equal |AP|=2, |BP|=3.
Find the maximal value of CP.
007.
Given some mxn table, and some numbers in its fields. You are allowed
to change the sign in one row or one column simultaneously.
Prove that You can obtain a table, with nonnegative sums over each row
and over each column.
008.
Given n points, some of them connected by non-intersecting segments.
You can reach every point from every one, moving along the
segments, and there is no couple, connected by two different ways.
Prove that the total number of the segments is (n-1).
009.
Given a, b, p -arbitrary integers.
Prove that there always exist relatively prime (i.e. that have no
common divisor) k and l, that (ak + bl) is divisible by p.
010.
Nicholas and Peter are dividing (2n+1) nuts. Each wants to get more.
Three ways for that were suggested. (Each consist of three stages.)
First two stages are common.
1 stage:
Peter divides nuts onto 2 heaps,
each contain not less than 2 nuts.
2 stage:
Nicholas divides both heaps onto 2 heaps,
each contain not less than 1 nut.
3 stage:
1 way:
Nicholas takes the biggest and the least heaps.
2 way:
Nicholas takes two middle size heaps.
3 way:
Nicholas takes either the biggest and the least heaps
or two middle size heaps, but gives one nut to the
Peter for the right of choice.
Find the most and the least profitable method for the Nicholas.
011.
Prove that for three arbitrary infinite sequences, of natural numbers
Given 120 unit squares arbitrarily situated in the 20x25 rectangle.
Prove that You can place a circle with the unit
diameter without intersecting any of the squares.
Given points A' ,B' ,C' ,D', on the continuation of the [AB], [BC], [CD], [DA]
sides of the convex quadrangle ABCD, such, that the following pairs of
vectors are equal:
[BB'[=[AB[,
[CC'[=[BC[,
[DD'[=[CD[,
[AA'[=[DA[.
Prove that the quadrangle A'B'C'D' area is five times more than the
quadrangle ABCD area.
014.
Given the circumference s and the straight line l, passing through the
centre O of s. Another circumference s' passes through the point O and
has its centre on the l.
Describe the set of the points M, where the
common tangent of s and s' touches s'.
015.
Given positive integers
a1,a2,...,a99,a100.
It is known, that
Prove that there are no integers a,b,c,d such that the polynomial
ax3+bx2+cx+d equals 1 at x=19, and equals
2 at x=62.
017.
Given a nxn table, where n is odd. There is either 1 or -1 in its every
field. A product of the numbers in the column is written under every
column. A product of the numbers in the row is written to the right of
every row.
Prove that the sum of 2n products doesn't equal to 0.
018.
Given two sides of the triangle.
Build that triangle, if medians to those sides are orthogonal.
019.
Given a quartet of positive numbers a,b,c,d, and is known, that abcd=1.
Prove that
a2+b2+c2+d2+ab+ac+ad+bc+bd+dc>=10
020.
Given right pentagon ABCDE. M is an arbitrary point inside ABCDE or on
its side. Let the distances |MA|, |MB|, ... , |ME| be renumerated and
denoted with
r1<=r2<=r3<=r4<=r5.
Find all the positions of the M, giving r3
the minimal possible value.
Find all the positions of the M, giving r3
the maximal possible value.
021.
Given 1962 -digit number. It is divisible by 9. Let x be the sum of its
digits. Let the sum of the digits of x be y. Let the sum of the digits
of y be z.
Find z.
022.
The M point is a middle of a isosceles triangle base [AC]. [MH] is
orthogonal to [BC] side. Point P is the middle of the segment [MH].
Prove that [AH] is orthogonal to [BP].
023.
What maximal area can have a triangle if its sides a,b,c satisfy inequality
0<=a<=1<=b<=2<=c<=3?
024.
Given x,y,z, three different integers.
Prove that ((x-y)5+(y-z)5+(z-x)5) is
divisible by 5(x-y)(y-z)(z-x).
025.
Given a0, a1, ... , an.
It is known that
a0=an=0;
ak-1-2ak+ak+1>=0
for all k = 1, 2, ... , k-1.
Prove that all the numbers are nonnegative.
026.
Given positive numbers a1, a2, ..., am;
b1, b2, ..., bn.
Is known that
a1+a2+...+am=b1+b2+...+bn.
Prove that You can fill an empty table with m rows and n columns with
no more than (m+n-1) positive number in such a way, that for all i,j
the sum of the numbers in the i-th row will equal to ai, and the sum
of the numbers in the j-th column -- to bj.
Given 5 circumferences, every four of them have a common point.
Prove that there exists a point that belongs to all five circumferences.
028.
Eight men had participated in the chess tournament. (Each meets each;
draws are allowed, giving 1/2 of pont; winner gets 1.) Everyone has
different number of points. The second one has got as many points as
the four weakest participants together.
What was the result of the play between the third prizer and
the chess-player that have occupied the seventh place?
029.
a)
Each diagonal of the quadrangle halves its area.
Prove that it is a parallelogram.
b)
Three main diagonals of the hexagon halve its area.
Prove that they intersect in one point.
030.
Natural numbers a and b are relatively prime.
Prove that the greatest common divisor of (a+b) and
(a2+b2) is either 1 or 2.
031.
Given two fixed points A and B .The point M runs along the
circumference containing A and B. K is the middle of the segment [MB].
[KP] is a perpendicular to the line (MA).
a)
Prove that all the possible lines (KP) pass through one point.
b)
Find the set of all the possible points P.
032.
Given equilateral triangle with the side l.
What is the minimal length d of a brush (segment), that will paint all
the triangle, if its ends are moving along the sides of the triangle.
033.
A chess-board 6x6 is tiled with the 2x1 dominos.
Prove that You can cut the board onto two parts by
a straight line that does not cut dominos.
034.
Given n different positive numbers
a1,a2,...,an.
We construct all the possible sums (from 1 to n terms).
Prove that among those sums there are at least n(n+1)/2 different ones.
035.
Given a triangle ABC. We build two angle bisectors in the corners A and
B. Than we build two lines parallel to those ones through the point C.
D and E are intersections of those lines with the bisectors. It happens,
that (DE) line is parallel to (AB).
Prove that the triangle is isosceles.
036.
Given the endless arithmetic progression with the positive integer
members. One of those is an exact square.
Prove that the progression
contain the infinite number of the exact squares.
037.
Given right 45-angle. Can You mark its corners with the digits
{0,1,...,9} in such a way, that for every pair of digits there would be
a side with both ends marked with those digits?
038.
Find such real p, q, a, b, that for all x an equality is held:
(2x-1)20 - (ax+b)20 =
(x2+px+q)10.
039.
On the ends of the diameter two "1"s are written. Each of the
semicircles is divided onto two parts and the sum of the numbers of its
ends (i.e. "2") is written at the middle point.
Then every of the four
arcs is halved and in its middle the sum of the numbers on its ends is
written.
Find the total sum of the numbers on the circumference after n steps.
040.
Given an isosceles triangle. Find the set of the points inside the
triangle such, that the distance from that point to the base equals to
the geometric mean of the distances to the sides.
The two heights in the triangle are not less than the respective sides.
Find the angles.
042.
Prove that for no natural m a number m(m+1) is a power of an integer.
043.
Given 1000000000 first natural numbers. We change each number with the
sum of its digits an repeat this procedure until there will remain
1000000000 one digit numbers.
Is there more "1"-s or "2"-s?
044.
Given an arbitrary set of 2k+1 integers:
{a1,a2,...,a2k+1}.
We make a new set:
{(a1+a2)/2, (a2+a3)/2,
(a2k+a2k+1)/2, (a2k+1+a1)/2};
and a new one, according to the same rule, and so on...
Prove that if we obtain integers only, the initial set consisted of
equal integers only.
045.
a)
Given a convex hexagon ABCDEF with all the equal angles.
Prove that |AB|-|DE| = |EF|-|BC| = |CD|-|FA|.
b)
The opposite problem: Prove that it is possible to build a convex
hexagon with equal angles of six segments
a1,a2,...,a6,
whose lengths satisfy the condition
a1-a4 =
a5-a2 =
a3-a6.
046.
Find integer solutions (x,y) of the equation (1964 times "sqrt"):
sqrt(x+sqrt(x+sqrt(....(x+sqrt(x))....)))=y.
047.
Four perpendiculars are drawn from the vertices of a convex quadrangle
to its diagonals.
Prove that their bases make a quadrangle similar to the given one.
048.
Find all the natural n such that n! is not divisible by n2.
049.
A honeybug crawls along the honeycombs with the unite length of their
hexagons. He has moved from the node A to the node B along the shortest
possible trajectory.
Prove that the half of his way he moved in one direction.
050.
The quadrangle ABCD is outscribed around the circle with the centre O.
Prove that the sum of AOB and COD angles equals 180 degrees.
051.
Given natural a,b,n. It is known, that for every natural k (k<>b) the
number a-kn is divisible by b-k.
Prove that a=bn.
052.
Given an expression x1:x2:...:xn
( : means division).
We can put the braces as we want.
How many expressions can we obtain?
053.
We have to divide a cube onto k non-overlapping tetrahedrons.
For what smallest k is it possible?
054.
Find the smallest exact square with last digit not 0, such that after
deleting its last two digits we shall obtain another exact square.
055.
Let ABCD be an outscribed trapezoid; E is a point of its diagonals
intersection; r1,r2,r3,r4 --
the radiuses of the circles
inscribed in the triangles ABE, BCE, CDE, DAE respectively.
Prove that
1/(r1)+1/(r3) = 1/(r2)+1/(r4).
a)
Each of the numbers x1,x2,...,xn
can be 1, 0, or -1.
What is the minimal possible value of the sum of all products of
couples of those numbers.
b)
Each absolute value of the numbers
x1,x2,...,xn
is less than 1.
What is the minimal possible value of the sum of all
products of couples of those numbers.
057.
Given a board 3x3 and 9 cards with some numbers (known to the
players). Two players, in turn, put those cards on the board.
The first wins if the sum of the numbers in the first and the
third row is greater than in the first and the third column.
Prove that it doesn't matter what numbers are on the cards,
but if the first plays the best way, the second can not win.
058.
A circle is outscribed around the triangle ABC. Chords, from the middle
of the arc AC to the middles of the arcs AB and BC, intersect sides
[AB] and [BC] in the points D and E.
Prove that (DE) is parallel to
(AC) and passes through the centre of the inscribed circle.
059.
A bus ticket is considered to be lucky if the sum of the first three
digits equals to the sum of the last three (6 digits in Russian buses).
Prove that the sum of all the lucky numbers is divisible by 13.
060.
There is a lighthouse on a small island. Its lamp enlights a segment of
a sea to the distance a. The light is turning uniformly, and the end of
the segment moves with the speed v.
Prove that a ship, whose speed doesn't exceed v/8 cannot arrive to the
island without being enlightened.
061.
A society created in the help to the police contains 100 man exactly.
Every evening 3 men are on duty.
Prove that You can not organise
duties in such a way, that every couple will meet on duty once exactly.
062.
What is the maximal possible length of the segment, being cut out by
the sides of the triangle on the tangent to the inscribed circle, being
drawn parallel to the base, if the triangle's perimeter equals 2p?
063.
Given n2 numbers xi,j (i,j=1,2,...,n)
satisfying the system of n3 equations:
xi,j+xj,k+xk,i=0 (i,j,k = 1,...,n).
Prove that there exist such numbers
a1,a2,...,an;
that for all i,j=1,...n xi,j=ai-aj.
064.
Is it possible to pose 1965 points inside the unit square to make every
rectangle with the square 0.005 and sides parallel to the side of the
square to contain at least one of those points?
065.
Quasi-rounding is a substitution one of the two closest integers
instead of the given number.
Given n numbers. Prove that You can quasi-round them in such a way,
that a sum of every subset of quasi-rounded numbers will deviate from
the sum of the same subset of initial numbers not greater than (n+1)/4.
066.
The tourist has come to the Moscow by train. All-day-long he wandered
randomly through the streets. Than he had a supper in the cafe on the
square and decided to return to the station only through the streets
that he has passed an odd number of times.
Prove that he is always able to do that.
067.
a)
A certain committee has gathered 40 times. There were 10 members on
every meeting. Not a single couple has met on the meetings twice.
Prove that there were no less then 60 members in the committee.
b)
Prove that You can not construct more then 30 subcommittees
of 5 members from the committee of 25 members, with no
couple of subcommittees having more than one common member.
068.
Given two relatively prime numbers p>0 and q>0.
An integer n is called "good" if we can represent it as n = px + qy
with nonnegative integers x and y, and "bad" in the opposite case.
a)
Prove that there exist integer c such that in a pair
{n, c-n} always one is "good" and one is "bad".
b)
How many there exist "bad" numbers.?
069.
An airplane-spy flies on the circumference with the centre A and radius
10 km. Its speed is 1000 km/h. At the certain moment a rocket with the
same speed starts from the point A and moves remaining on one line with
the plain and A.
What time is necessary for it to hit the plane?
/* remember, no advanced math. were known to the participants -VAP */
070.
Prove that the sum of the lengths of the polyhedron edges
exceeds its tripled diameter (distance between two farest vertices).
071.
On the surface of the planet lives one inhabitant, that can move with
the speed not greater than u. A spaceship approaches to the planet with
its speed v.
Prove that if v/u > 10 , the spaceship can find the
inhabitant, even it is trying to hide.
There is exactly one astronomer on every planet of a certain system. He
watches the closest planet. The number of the planets is odd and all of
the distances are different.
Prove that there one planet being not watched.
073.
a)
Points B and C are inside the segment [AD]. |AB|=|CD|.
Prove that for all of the points P on the plane holds inequality
|PA|+|PD|>|PB|+|PC|.
b)
Given four points A,B,C,D on the plane. For all of the points P
on the plane holds inequality |PA|+|PD| > |PB|+|PC|.
Prove that points B and C are inside the segment [AD] and |AB|=|CD|.
074.
Can both (x2+y) and (y2+x)
be exact squares for natural x and y?
075.
a)
Pupils of the 8-th form are standing in a row. There is the pupil
of the 7-th form in before each, and he is smaller (in height) than
the elder.
Prove that if You will sort the pupils in each of rows
with respect to their height, every 8-former will still be taller
than the 7-former standing before him.
b)
An infantry detachment soldiers stand in the rectangle, being
arranged in columns with respect to their height.
Prove that if You
rearrange them with respect to their height in every separate row,
they will still be staying with respect to their height in columns.
076.
A rectangle ABCD is drawn on the cross-lined paper with its sides
laying on the lines, and |AD| is k times more than |AB| (k is an
integer). All the shortest paths from A to C coming along the lines are
considered.
Prove that the number of those with the first link on [AD]
is k times more then of those with the first link on [AB].
Prove that in the sum s=+-a1+-a2+-...+-an
You can choose appropriate signs to make 0<=s<=a1.
078.
Prove that You can always pose a circle of radius S/P
inside a convex polygon with the perimeter P and area S.
079.
For three arbitrary crossroads A,B,C in a certain city there exist a
way from A to B not coming through C.
Prove that for every couple of
the crossroads there exist at least two non-intersecting ways
connecting them. (there are at least two crossroads in the city)
080.
Given a triangle ABC. Consider all the tetrahedrons PABC
with PH -- the smallest of all tetrahedron's heights.
Describe the set of all possible points H.
081.
Given 100 points on the plain.
Prove that You can cover them with a
family of circles with the sum of their diameters less than 100 and the
distance between any two of the circles more than one.
082.
The distance from A to B is d kilometres. A plain flying with the
constant speed in the constant direction along and over the line (AB)
is being watched from those points. Observers have reported that the
angle to the plain from the point A has changed by \alpha degrees and
from B --- by \beta degrees within one second.
What can be the minimal speed of the plain?
083.
20 Numbers are written on the board: 1, 2, ... ,20. Two players are
putting signs before the numbers in turn (+ or -). The first wants to
obtain the minimal possible absolute value of the sum.
What is the maximal value of the absolute value of the sum that can be
achieved by the second player?
"Where is the beginning of that end, that ends the beginning?"
Kozma Prutkov.
"If You see a title 'buffalo' on the elephant's cage
-- don't believe to Your eyes!"
Kozma Prutkov.
The name has been changed -- the numeration was restarted.
a)
The maximal height |AH| of the acute-angled triangle ABC equals the
median |BM|.
Prove that the angle ABC isn't greater than 60 degrees.
b)
The height |AH| of the acute-angled triangle ABC equals the median
|BM| and bisectrix |CD|.
Prove that the angle ABC is equilateral.
085.
a)
The digits of a natural number were rearranged.
Prove that the sum
of given and obtained numbers can't equal 999...9 (1967 of nines).
b)
The digits of a natural number were rearranged.
Prove that if the
sum of the given and obtained numbers equals 1010, than the
given number was divisible by 10.
086.
a)
A lamp of a lighthouse enlights an angle of 90 degrees.
Prove that You can turn the lamps of four arbitrary
posed lighthouses so, that all the plane will be enlightened.
b)
There are eight lamps in eight points of the space. Each can
enlighten an octant (three-faced space angle with three mutually
orthogonal edges).
Prove that You can turn them so, that all the space will be enlightened.
087.
a)
Can You pose the numbers 0,1,...,9 on the circumference in such a
way, that the difference between every two neighbours would be
either 3 or 4 or 5?
b)
The same question, but about the numbers 0,1,...,13.
088.
Prove that there exists a number divisible by 51000
not containing a single zero in its decimal notation.
089.
Find all the integers x,y satisfying equation
x2+x=y4+y3+y2+y.
090.
In the sequence of the natural (i.e. positive integers) numbers every
member from the third equals the absolute value of the difference of
the two previous.
What is the maximal possible length of such a
sequence, if every member is less or equal to 1967?
091.
"KING-THE SUICIDER"
Given a chess-board 1000x1000, 499 white castles and a black king.
Prove that it does not matter neither the initial situation nor the way
white plays, but the king can always enter under the check in a finite
number of moves.
092.
Three vertices KLM of the rhombus (diamond) KLMN lays on
the sides [AB], [BC] and [CD] of the given unit square.
Find the area of the set of all the possible vertices N.
093.
Given natural number k with a property "if n is divisible by k, than
the number, obtained from n by reversing the order of its digits is
also divisible by k".
Prove that the k is a divisor of 99.
The second competition -- Leningrad, 1968.
form first day second day
8 094 095 096 097 098 | 105a 106 107 108 109
9 099 100 101 097 102 | 110 111 105a 108 109
10 103 095 104 097 096 | 105b 112 113 114 109
094.
Given an octagon with the equal angles. The lengths of all the sides
are integers.
Prove that the opposite sides are equal in pairs.
095.
What is greater, 3111 or 1714?
/* calculators were not available - VAP */
096.
The circumference with the radius 100cm is drawn on the cross-lined
paper with the side of the squares 1cm. It neither comes through the
vertices of the squares, nor touches the lines.
How many squares can
it pass through?
097.
Some students on the faculty speak several languages and some - Russian
only. 50 of them know English, 50 -- French and 50 -- Spanish.
Prove that it is possible to divide them onto 5 groups, not necessary equal,
to get 10 of them knowing English, 10 -- French and 10 -- Spanish in
each group.
Given two acute-angled triangles ABC and A'B'C' with the points O and
O' inside. Three pairs of the perpendiculars are drawn:
[OA1] to the side [BC], [O'A'1] to the side [B'C'],
[OB1] to the side [AC], [O'B'1] to the side [A'C'],
[OC1] to the side [AB], [O'C'1] to the side [A'B'];
it is known that
[OA1] is parallel to the [O'A'],
[OB1] is parallel to the [O'B'],
[OC1] is parallel to the [O'C']
and the following products are equal:
|OA1|*|O'A'| = |OB1|*|O'B'| = |OC1|*|O'C'|.
Prove that
[O'A'1] is parallel to the [OA],
[O'B'1] is parallel to the [OB],
[O'C'1] is parallel to the [OC]
and
|O'A'1|*|OA| = |O'B'1|*|OB| = |O'C'1|*|OC|.
102.
Prove that You can represent an arbitrary number not exceeding n!
(n-factorial; n!= 1*2*3*...*n) as a sum of k different numbers (k<=n)
that are divisors of n!.
103.
Given a triangle ABC, point D on [AB], E on [AC]; |AD| = |DE| = |AC| ,
|BD| = |AE| , DE is parallel to BC.
Prove that the length |BD| equals
to the side of a right decagon (ten-angle) inscribed in a circle with
the radius R=|AC|.
104.
Three spheres are built so that the edges [AB], [BC], [AD] of
the tetrahedron ABCD are their respective diameters.
Prove that the spheres cover all the tetrahedron.
The fields of the square table 4x4 are filled with the "+" or
"-" signs. You are allowed to change the signs simultaneously in
the whole row, column, or diagonal to the opposite sign. That means, for
example, that You can change the sign in the corner square, because
it makes a diagonal itself.
Prove that You will never manage to obtain a table containing pluses only.
b)
The fields of the square table 8x8 are filled with the "+" or signs
except one non-corner field with "-". You are allowed to change the
signs simultaneously in the whole row, column, or diagonal to the
opposite sign. That means, for example, that You can change the sign in
the corner field, because it makes a diagonal itself.
Prove that You
will never manage to obtain a table containing pluses only.
106.
Medians divide the triangle onto 6 smaller ones. 4 of the
circles inscribed in those small ones are equal.
Prove that the triangle is equilateral.
107.
Prove that the equation x2 + x + 1 = py has solution
(x,y) for the infinite number of simple p.
108.
Each of the 9 referees on the figure skating championship estimates the
program of 20 sportsmen by assigning him a place (from 1 to 20). The
winner is determined by adding those numbers. (The less is the sum -
the higher is the final place).
It was found, that for the each sportsman, the difference of the
places, received from the different referees was not greater than 3.
What can be the maximal sum for the winner?
109.
Two finite sequences a1,a2,...,an;
b1,b2,...,bn
are just rearranged sequence 1, 1/2, ... , 1/n.
a1+b1>=a2+b2>=...>=an+bn.
Prove that for every m (1<=m<=n) am+an>=4/m.
110.
There is scales on the teacher's table. There is a set of weighs on the
scales, and there are some pupils' names (may be more than one) on the
every weigh. A pupil entering the classroom moves all the weight with
his name to another side of the scales.
Prove that You can let in such
a subset of the pupils, that the scales will change its position.
111.
The city is a rectangle divided onto squares by m streets coming from
the West to the East and n streets coming from the North to the South.
There are militioners (policemen) on the streets but not on the
crossroads. They watch the certain automobile, moving along the closed
route, marking the time and the direction of its movement. Its trace is
not known in advance, but they know, that it will not pass over the
same segment of the way twice.
What is the minimal number of the militioners providing the unique
determination of the route according to their reports?
112.
The circle inscribed in the triangle ABC touches the side [AC] in the
point K.
Prove that the line connecting the middle of the [AC] side
with the centre of the circle halves the [BK] segment.
113.
The sequence a1,a2,...,an satisfies the following conditions:
a1=0, |a2|=|a1+1|, ..., |an|=|an-1+1|.
Prove that (a1+a2+...+an)/n>=-1/2.
114.
Given a quadrangle ABCD. The lengths of all its sides and diagonals are
the rational numbers. Let O be the point of its diagonals intersection.
Prove that |AO| - the length of the [AO] segment is also rational.
The third competition -- Kiev, 1969.
form first day second day
8 115 116 117 | 122 123 124a
9 118 119 115 | 124 125 126
10 119 120 121 | 125 126 128
115.
The point E lies on the base [AD] of the trapezoid ABCD. The triangles
ABE, BCE and CDE perimeters are equal.
Prove that |BC| = |AD|/2.
116.
There is a wolf in the centre of a square field, and four dogs in the
corners. The wolf can easily kill one dog, but two dogs can kill the
wolf. The wolf can run all over the field, and the dogs -- along the
fence (border) only.
Prove that if the dog's speed is 1.5 times more
than the wolf's, than the dogs can prevent the wolf escaping.
117.
Given a finite sequence of 0's and 1's with two properties:
if you chose five sequential digits in one place and in
the second place, those will be two different binary numbers.
(Some last digits of the first number may be included as
the first digits in the second.)
if You add 0 or 1 either from the left or from the right
side, the previous property will not be held.
Prove that the first four digits of that sequence coincide with the
last four.
118.
Given positive numbers a,b,c,d.
Prove that the set of inequalities
a+b<c+d;
(a+b)(c+d)<ab+cd;
(a+b)cd<ab(c+d)
contain at least one wrong.
119.
For what minimal natural a the polynomial ax2 + bx + c with the
integer c and b has two different positive roots both less than one.
120.
Given natural n. Consider all the fractions 1/(pq), where p and q are
relatively prime;
0<p<q<=n ; p+q>n.
Prove that the sum of all such a fractions equals to 1/2.
121.
Given n points in the three dimensional space such, that the arbitrary
triangle with the vertices in three of those points contains an angle
greater than 120 degrees.
Prove that You can rearrange them to make a polyline (unclosed) with
all the angles between the sequent links greater than 120 degrees.
122.
Find four different three-digit decimal numbers starting with the same
digit, such that their sum is divisible by three of them.
123.
Every city in the certain state is connected by airlines with no more
than with three other ones, but one can get from every city to every
other city changing a plane once only or directly.
What is the maximal possible number of the cities?
124.
Given a pentagon with all equal sides.
a)
Prove that there exist such a point on the maximal diagonal,
that every side is seen from it inside a right angle.
/* I mean that the side AB is seen from the point C inside an
arbitrary angle that is greater or equal than angle ACB. - VAP */
b)
Prove that the circles build on its sides as on the diameters cannot
cover the pentagon entirely.
125.
Given an equation x3 + ?x2 + ?x + ? = 0. First player substitutes
an integer on the place of one of the interrogative marks, than the
same do the second with one of the two remained marks, and, finally,
the first puts the integer instead of the last mark.
Explain how can the first provide the existence of three integer roots
in the obtained equation. (The roots may coincide.)
126.
20 football teams participate in the championship. What minimal number
of the games should be played to provide the property: from the three
arbitrary teams we can find at least on pair that have already met in
the championship.
127.
Let hk be an apothem of the right k-angle inscribed into a circle with
radius R.
Prove that (n + 1)hn+1 - nhn > R.
128.
Prove that for the arbitrary positive a1, a2, ... , an
the following inequality is held
form first day second day
8 129 130 131 132 133a
9 134 135 133b 136 137
10 138 139 133b 136 140 | 141 142 143
129.
Given a circle, its diameter [AB] and a point C on it.
Build (with the
help of compasses and ruler) two points X and Y, that are symmetric
with respect to (AB) line, such that (YC) is orthogonal to (XA).
130.
The product of three positive numbers equals to one, their
sum is strictly greater than the sum of the inverse numbers.
Prove that one and only one of them is greater than one.
131.
How many sides of the convex polygon can equal its longest diagonal?
132.
The digits of the 17-digit number are rearranged in the reverse order.
Prove that at list one digit of the sum of the new and the initial
number is even.
133.
a)
A castle is equilateral triangle with the side of 100 metres. It is
divided onto 100 triangle rooms. Each wall between the rooms is 10
metres long and contain one door. You are inside and are allowed to
pass through every door not more than once.
Prove that You can visit not more than 91 room (not exiting the castle).
b)
Every side of the triangle is divided onto k parts by the lines
parallel to the sides. And the triangle is divided onto k2
small triangles. Let us call the "chain" such a sequence of
triangles, that every triangle in it is included only once,
and the consecutive triangles have the common side.
What is the greatest possible number of the triangles in the chain?
134.
Given five segments. It is possible to build
a triangle of every subset of three of them.
Prove that at least one of those triangles is acute-angled.
135.
The bisector [AD], the median [BM] and the height [CH] of
the acute-angled triangle ABC intersect in one point.
Prove that the angle BAC is greater than 45 degrees.
136.
Given five n-digit binary numbers. For each two numbers
their digits coincide exactly on m places. There is no
place with the common digit for all the five numbers.
Prove that 2/5 <= m/n <= 3/5.
137.
Prove that from every set of 200 integers You can choose a
subset of 100 with the total sum divisible by 100.
138.
Given triangle ABC, middle M of the side [BC], the centre O of the
inscribed circle. The line (MO) crosses the height AH in the point E.
Prove that the distance |AE| equals the inscribed circle radius.
139.
Prove that for every natural number k there exists an infinite set of
such natural numbers t, that the decimal notation of t does not contain
zeroes and the sums of the digits of the numbers t and kt are equal.
140.
Two equal rectangles are intersecting in 8 points.
Prove that the
common part area is greater than the half of the rectangle's area.
141.
All the 5-digit numbers from 11111 to 99999 are written on
the cards. Those cards lies in a line in an arbitrary order.
Prove that the resulting 444445-digit number is not a power of two.
142.
All natural numbers containing not more than n digits are divided onto
two groups. The first contains the numbers with the even sum of the
digits, the second -- with the odd sum.
Prove that if 0<k<n than
the sum of the k-th powers of the numbers in the first group equals to
the sum of the k-th powers of the numbers in the second group.
143.
The vertices of the right n-angle are marked with some colours
(each vertex -- with one colour) in such a way, that the vertices
of one colour represent the right polygon.
Prove that there are two equal ones among the smaller polygons.
The 5-th competition -- Riga, 1971.
form first day second day
8 144 145a 146a 147 | 152ab 153 154
9 144 145a 148 147 146b | 156abc152c 155
10 149 145b 150 147 151b | 156 157 158
144.
Prove that for every natural n there exists a number, containing only
digits "1" and "2" in its decimal notation, that is divisible by 2n
( n-th power of two ).
145.
a)
Given a triangle A1A2A3 and the points
B1 and D2 on the side [A1A2],
B2 and D3 on the side [A2A3],
B3 and D1 on the side [A3A1].
If You build parallelograms A1B1C1D1,
A2B2C2D2
and A3B3C3D3, the lines
(A1C1), (A2C2) and (A3C3), will
cross in one point O.
Prove that if |A1B1| = |A2D2| and
|A2B2| = |A3D3|, than |A3B3| = |A1D1|.
b)
Given a convex polygon A1A2 ... An and the points
B1 and D2 on the side [A1A2],
B2 and D3 on the side [A2A3],
........
Bn and D1 on the side [AnA1].
If You build parallelograms A1B1C1D1,
A2B2C2D2 ... ,
AnBnCnDn, the lines (A1C1), (A2C2), ..., (AnCn), will
cross in one point O.
Prove that
|A1B1|*|A2B2|*...*|AnBn| =
|A1D1|*|A2D2|*...*|AnDn|.
146.
a)
A game for two.
The first player writes two rows of ten numbers
each, the second under the first. He should provide the following
property: if number b is written under a, and d -- under c, then a
+ d = b + c.
The second player has to determine all the numbers. He
is allowed to ask the questions like "What number is written in the
x place in the y row?"
What is the minimal number of the questions
asked by the second player before he founds out all the numbers?
b)
There was a table mxn on the blackboard with the property: if You
chose two rows and two columns, then the sum of the numbers in the
two opposite vertices of the rectangles formed by those lines equals
the sum of the numbers in two another vertices. Some of the numbers
are cleaned. but it is still possible to restore all the table.
What is the minimal possible number of the remaining numbers?
147.
Given an unite square and some circles inside. Radius of each circle is
less than 0.001, and there is no couple of points belonging to the
different circles with the distance between them 0.001 exactly.
Prove that the area, covered by the circles is not greater than 0.34.
148.
The volumes of the water containing in each of three big enough
containers are integers. You are allowed only to relocate some times
from one container to another the same volume of the water, that the
destination already contains.
Prove that You are able to discharge one of the containers.
149.
Prove that if the numbers p1, p2, q1, q2 satisfy the condition
(q1 - q2)2 + (p1 - p2)(p1q2 - p2q1) < 0,
then the square polynomials
x2 + p1x + q1 and
x2 + p2x + q2
have real roots, and between the roots of each there is a root of another one.
150.
The projections of the body on two planes are circles.
Prove that they have the same radius.
151.
Some numbers are written along the ring. If inequality (a-d)(b-c) < 0
is held for the four arbitrary numbers in sequence a,b,c,d, You have to
change the numbers b and c places.
Prove that You will have to do this operation finite number of times.
152.
a)
Prove that the line dividing the triangle onto two polygons with
equal perimeters and equal areas passes through the centre of the
inscribed circle.
b)
Prove the same statement for the arbitrary polygon outscribed around
the circle.
c)
Prove that all the lines halving its perimeter and area
simultaneously, intersect in one point.
153.
Given 25 different positive numbers.
Prove that You can choose two of
them such, that none of the other numbers equals neither to the sum nor
to the difference between the chosen numbers.
154.
a)
The vertex A1 of the right 12-angle (dodecagon) A1A2...A12 is
marked with "-" and all the rest -- with "+". You are allowed to
change the sign simultaneously in the 6 vertices in succession.
Prove that is impossible to obtain dodecagon with A2 marked with
"-" and the rest of the vertices -- with "+".
b)
Prove the same statement if it is allowed to change the signs not in
six, but in four vertices in succession.
c)
Prove the same statement if it is allowed to change the signs in
three vertices in succession.
155.
N unit squares on the infinite sheet of cross-lined paper are painted
with black colour.
Prove that You can cut out the finite number of
square pieces and satisfy two conditions
all the black squares are contained in those pieces.
the area of black squares is not less than 1/5 and not greater than
4/5 of every piece area.
The task for the tenth form was as follows:
"You are given three serious problems. Try to investigate at least one,
but to obtain as many results, as You can. At the end of Your work
make a sort of resume, showing the main proved facts, challenged
examples and the hypotheses that seem to be true ..."
/* this form of competition was never repeated later - it had
required too much efforts from those who checked the works */
156.
A cube with the edge of length n is divided onto n3 unit ones.
Let us choose some of them and draw three lines parallel to the edges through
their centres.
What is the least possible number of the chosen small
cubes necessary to make those lines cross all the smaller cubes?
a)
Find the answer for the small n (n = 2,3,4).
b)
Try to find the answer for n = 10.
c)
If You can not solve the general problem, try to estimate that value
from the upper and lower side.
d)
Note, that You can reformulate the problem in such a way:
Consider all the triples (x1,x2,x3), where xi can be one of the
integers 1,2,...,n.
What is the minimal number of the triples
necessary to provide the property: for each of the triples there
exist the chosen one, that differs only in one coordinate.
Try to
find the answer for the situation with more than three coordinates,
for example, with four.
157.
a)
Consider the function f(x,y) = x2 + xy + y2.
Prove that for the every point (x,y) there exist such
integers (m,n), that f((x-m),(y-n)) <= 1/2.
b)
Let us denote with g(x,y) the least possible value of the
f((x-m),(y-n)) for all the integers m,n. The statement a)
was equal to the fact g(x,y) <= 1/2.
Prove that in fact, g(x,y) <= 1/3.
Find all the points (x,y), where g(x,y)=1/3.
c)
Consider function fa(x,y) = x2 + axy + y2 (0 <= a <= 2).
Find
any c such that ga(x,y) <= c.
Try to obtain the closest estimation.
158.
1 2 1 2
| | | | The switch with two inputs and two
v v v v outputs can be in one of two
+-+---+-+ +-+---+-+ different positions. In the left
| | | | | | | | part of the picture a) the first input
| \ / | | | | | is connected with the second output
| X | | | | | and we can denote this as
| / \ | | | | | 1 2
| | | | | | | | | | and the position in the right
+-+---+-+ +-+---+-+ V V part of the picture 1 2
| | | | 2 1 will be denoted as | |
V V V V V V
1 2 1 2 1 2
fig.a)
1 2 3 The scheme on the picture b) is universal
| | | in that sense that changing the state of
V V | the element switches You can obtain all
+-+---+-+ | the six connections, i.e.
| | |
| | | 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
+-+---+-+ | | | | | | | | | | | | | | | | | | |
| | | V V V V V V V V V V V V V V V V V V
| V V 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1
| +-+---+-+
| | | (Check it. Note, that the total number of the
| | | states is 2 * 2 * 2 = 8, because each element
| +-+---+-+ can be in two positions.)
| | |
V V |
+-+---+-+ |
| | |
| | |
+-+---+-+ |
| | |
V V V
1 2 3
fig.b)
a)
Try to build the universal scheme for 4 inputs and 4 outputs,
that can provide all of 24 possible connections.
b)
What is the minimal number of the element switches for such a
scheme?
Let us call a scheme with n inputs and n outputs n-universal, if it can
provide all n! possible connections of n inputs with n outputs.
c)
Here is the scheme (picture c) with 8 inputs and 8 outputs, where A
and B are 4-universal.
Prove that it is 8-universal.
1 2 3 4 5 6 7 8
| | | | | | | |
V V V V V V V V V V V V V V V V
+-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+
| | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | |
+-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+
| \ | \ | \ | \ / | / | / | / |
| \ | \ | \ | \ / | / | / | / |
| \ | \ | \ | X | / | / | / |
......................................................................
|/ |/ |/ |/ \| \| \| \|
+-+--------+--------+--------+----+ +----+--------+--------+--------+-+
| | | |
| A | | B |
| | | |
+-+--------+--------+--------+----+ +----+--------+--------+--------+-+
|\ |\ |\ |\ /| /| /| /|
......................................................................
| / | / | / | / \ | \ | \ | \ |
| / | / | / | / \ | \ | \ | \ |
V V V V V V V V V V V V V V V V
+-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+
| | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | |
+-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+ +-+--+-+
| | | | | | | |
V V V V V V V V
1 2 3 4 5 6 7 8
fig.c)
d)
Estimate the upper and lower bound for the number of the element
switches in the n-universal scheme.
The 6-th competition -- Chelyabinsk, 1972.
form first day second day
8 159 160 161 | 166 167 168
9 162a 163 161 164 | 169 170 171
10 162b 163 165 164 | 166 172 173
159.
Given a rectangle ABCD, points M -- the middle of [AD] side, N -- the
middle of [BC] side. Let us take a point P on the continuation of the
[DC] segment over the point D. Let us denote the point of intersection
of lines (PM) and (AC) as Q.
Prove that the angles QNM and MNP are equal.
160.
Given 50 segments on the line. Prove that one of the following
statements is valid:
1. Some 8 segments have the common point.
2. Some 8 segments do not intersect each other.
161.
Find the maximal x such that the expression 427 + 41000 + 4x
is the exact square.
162.
a)
Let a,n,m be natural numbers, a > 1.
Prove that if (am + 1) is
divisible by (an + 1) than m is divisible by n.
b)
Let a,b,n,m be natural numbers, a>1, a and b are relatively prime.
Prove that if (am+bm) is divisible by
(an+bn) than m is
divisible by n.
163.
2
/ \
/ \
4 3
/ \ / \
16 5 9 4
/ \ / \ /\ / \
The triangle table is built according to the rule:
You put the natural number a>1 in the upper row, and
then You write under the number k from the left
side k2, and from the right side -- (k+1). For
example, if a = 2, You get the table on the picture.
Prove that all the numbers on
each particular line are different.
164.
Given several squares with the total area 1.
Prove that You can pose
them in the square of the area 2 without any intersections.
165.
Let O be the intersection point of the of the convex quadrangle ABCD
diagonals.
Prove that the line drawn through the points of
intersection of the medians of AOB and COD triangles is orthogonal to
the line drawn through the points of intersection of the heights of BOC
and AOD triangles.
166.
Each of the 9 straight lines divides the given square onto two
quadrangles with the areas related as 2:3.
Prove that there exist
three of them intersecting in one point.
167.
The 7-angle A1A2A3A4A5A6A7 is inscribed in a circle.
Prove that if the centre of the circle is inside the 7-angle, than the
sum of A1,A2 and A3 angles is less than 450 degrees.
168.
A game for two. One gives a digit and the second substitutes
it instead of a star in the following difference:
**** - **** =
Then the first gives the next digit, and so on 8 times.
The first wants to obtain the greatest possible difference,
the second -- the least. Prove that:
1. The first can operate in such a way that the difference would be
not less than 4000, not depending on the second's behaviour.
2. The second can operate in such a way that the difference would be
not greater than 4000, not depending on the first's behaviour.
169.
Let x,y be positive numbers, s -- the least of { x, (y+ 1/x), 1/y}.
What is the greatest possible value of s? To what x and y does it
correspond?
170.
The point O inside the convex polygon makes isosceles triangle with all
the pairs of its vertices.
Prove that O is the centre of the outscribed circle.
171.
Is it possible to put the numbers 0,1 or 2 in the unit squares of the
cross-lined paper 100x100 in such a way, that every rectangle 3x4 (and
4x3) would contain three zeros, four ones and five twos?
172.
Let the sum of positive numbers x1, x2, ... , xn be 1.
Let s be the greatest of the numbers
{x1/(1+x1), x2/(1+x1+x2),
... xn/(1+x1+...+xn)}.
What is the minimal possible s? What xi correspond it.
/* Derivatives were not included in the school plans in
Russia that time. They expected another solution. -VAP */
173.
One-round hockey tournament is finished (each plays with each one time,
the winner gets 2 points, looser -- 0, and 1 point for draw). For
arbitrary subgroup of teams there exists a team (may be from that
subgroup) that has got an odd number of points in the games with the
teams of the subgroup.
Prove that there was even number of the participants.
The 7-th competition -- Kishenew, 1973.
form first day second day
8 174a 175 176 | 182 183 184ab
9 174b 177 178 | 179 185 186 184c
10 180 177 181 | 187 188 184c
174.
Fourteen coins are submitted to the judge. An expert knows, that the
coins from number one to seven are false, and from 8 to 14 -- normal.
The judge is sure only that all the true coins have the same weight and
all the false coins weights equal each other, but are less then the
weight of the true coins. The expert has the scales without weights.
a)
The expert wants to prove, that the coins 1--7 are false.
How can he do it in three weighings?
b)
How can he prove, that the coins 1--7 are false and the coins 8--14
are true in three weighings?
175.
Prove that 9-digit number, that contains all the decimal digits except
zero and does not ends with 5 can not be exact square.
176.
Given n points, n > 4. Prove that You can connect them with arrows, in
such a way, that You can reach every point from every other point,
having passed through one or two arrows. (You can connect every pair
with one arrow only, and move along the arrow in one direction only.)
177.
Given an angle with the vertex O and a circle touching its sides in the
points A and B. A ray is drawn from the point A parallel to [OB). It
intersects with the circumference in the point C. The segment [OC]
intersects the circumference in the point E. The straight lines (AE)
and (OB) intersect in the point K.
Prove that |OK| = |KB|.
178.
The real numbers a,b,c satisfy the condition: for all x, such
that for -1 <= x <= 1, the inequality
| ax2 + bx + c | <= 1 is held.
Prove that for the same x
| cx2 + bx + a | <= 2.
179.
The tennis federation has assigned numbers to 1024 sportsmen,
participating in the tournament, according to their skill. (The tennis
federation uses the olympic system of tournaments. The looser in the
pair leaves, the winner meets with the winner of another pair. Thus, in
the second tour remains 512 participants, in the third -- 256, et.c.
The winner is determined after the tenth tour.) It comes out, that in
the play between the sportsmen whose numbers differ more than on 2
always win that whose number is less.
What is the greatest possible number of the winner?
180.
The square polynomial f(x)=ax2+bx+c is of such a sort,
that the equation f(x)=x does not have real roots.
Prove that the equation f(f(x)) does not have real roots also.
181.
n squares of the infinite cross-lined sheet of paper are painted
with black colour (others are white). Every move all the squares
of the sheet change their colour simultaneously. The square gets
the colour, that had the majority of three ones: the square itself,
its neighbour from the right side and its neighbour from the upper side.
a)
Prove that after the finite number of the moves all the black
squares will disappear.
b)
Prove that it will happen not later than on the n-th move.
182.
Three similar acute-angled triangles AC1B, BA1C and CB1A are built
on the outer side of the acute-angled triangle ABC. (Equal triples of
the angles are AB1C, ABC1, A1BC and BA1C, BAC1, B1AC.)
a)
Prove that the circumferences outscribed around the outer
triangles intersect in one point.
b)
Prove that the straight lines AA1, BB1 and CC1
intersect in the same point.
183.
N men are not acquainted each other. You need to introduce some of them
to some of them in such a way, that all the men will have different
number of the acquaintances.
Prove that it is possible for all N.
184.
The king have revised the chess-board 8x8 having visited all the fields
once only and returned to the starting point. When his trajectory was
drawn (the centres of the squares were connected with the straight
lines), a closed broken line without self-intersections appeared.
a)
Give an example that the king could make 28 steps parallel
the sides of the board only.
b)
Prove that he could not make less than 28 such a steps.
c)
What is the maximal and minimal length of the broken line
if the side of a field is 1?
185.
Given a triangle with a,b,c sides and with the area 1. a >= b >= c.
Prove that b2 >= 2.
186.
Given a convex n-angle with pairwise (mutually) non-parallel sides
/* who knows Russian -- a letter "r" had been broken on the
organising committee's typewriter - and it became
an inexhaustible source of jokes for some years */
and a point inside it. Prove that there are not more than n straight
lines coming through that point and halving the area of the n-angle.
187.
Prove that for every positive x1, x2, x3, x4, x5 holds inequality:
Given 4 points in three-dimensional space, not lying in one plane.
What is the number of such a parallelepipeds (bricks), that each point
is a vertex of each parallelepiped?
The 8-th competition -- Erevan, 1974.
form first day second day
8 189abc190 191 | 197 198 199 200a
9 190 192 189d 193 | 201 202 200b
10 194 195 196 193 | 203 204 200b
189.
a,b,c) Given some cards with either "-1" or "+1" written on the opposite
side. You are allowed to choose a triple of cards and ask about the
product of the three numbers on the cards.
What is the minimal number
of questions allowing to determine all the numbers on the cards ...
a)
for 30 cards,
b)
for 31 cards,
c)
for 32 cards.
(You should prove, that You cannot manage with less questions.)
d)
Fifty abovementioned cards are lying along the circumference. You
are allowed to ask about the product of three consecutive numbers
only. You need to determine the product af all the 50 numbers.
What is the minimal number of questions allowing to determine it?
190.
Among all the numbers representable as 36k - 5l (k and l are
natural numbers) find the smallest.
Prove that it is really the smallest.
191.
a)
Each of the side of the convex hexagon (6-angle) is longer
than 1. Does it necessary have a diagonal longer than 2?
b)
Each of the main diagonals of the convex hexagon is longer
than 2. Does it necessary have a side longer than 1?
192.
Given two circles with the radiuses R and r, touching each other from
the outer side. Consider all the trapezoids, such that its lateral
sides touch both circles, and its bases touch different circles.
Find the shortest possible lateral side.
193.
Given n vectors of unit length in the plane. The length of
their total sum is less than one.
Prove that You can
rearrange them to provide the property: for every k, k<= n,
the length of the sum of the first k vectors is less than 2.
Given a square ABCD. Points P and Q are in the sides [AB] and
[BC] respectively. |BP|=|BQ|. Let H be the base of the
perpendicular from the point B to the segment [PC].
Prove that the angle DHQ is a right one.
196.
Given some red and blue points. Some of them are connected by the
segments. Let us call "exclusive" the point, if its colour differs from
the colour of more than half of the connected points. Every move one
arbitrary "exclusive" point is repainted to the other colour.
Prove that after the finite number of moves there will remain no
"exclusive" points.
197.
Find all the natural n and k such that nn has k digits and kk has n
digits.
198.
Given points D and E on the legs [CA] and [CB], respectively, of the
isosceles right triangle. |CD| = |CE|. The extensions of the
perpendiculars from D and C to the line AE cross the hypotenuse AB in
the points K and L.
Prove that |KL| = |LB|.
199.
Two are playing the game "cats and rats" on the chess-board 8x8. The
first has one piece -- a rat, the second -- several pieces -- cats. All
the pieces have four available moves -- up, down, left, right -- to the
neighbour field, but the rat can also escape from the board if it is on
the boarder of the chess-board. If they appear on the same field -- the
rat is eaten. The players move in turn, but the second can move all the
cats in independent directions.
a)
Let there be two cats. The rat is on the interior field.
Is it possible to put the cats on such a fields on the border
that they will be able to catch the rat?
b)
Let there be three cats, but the rat moves twice during the
first turn. Prove that the rat can escape.
200.
a)
Prove that You can rearrange the numbers 1, 2, ... , 32 in
such a way, that for every couple of numbers none of the
numbers between them will equal their arithmetic mean.
b)
Can You rearrange the numbers 1, 2, ... , 100 in such a way,
that for every couple of numbers none of the numbers between
them will equal their arithmetic mean?
201.
Find all the three-digit numbers such that it equals to the arithmetic
mean of the six numbers obtained by rearranging its digits.
202.
Given a convex polygon. You can put no triangle with area 1 inside it.
Prove that You can put the polygon inside a triangle with the area 4.
203.
Given a function f(x) on the segment 0<=x<=1.
For all x,
f(x)>=0; f(1)=1.
For all the couples of {x1,x2} such, that all
the arguments are in the segment f(x1+x2)>=f(x1)+f(x2).
a)
Prove that for all x holds f(x) <= 2x.
b)
Is the inequality f(x) <= 1.9x valid?
204.
Given a triangle ABC with the are 1. Let A',B' and C' are
the middles of the sides [BC], [CA] and [AB] respectively.
What is the minimal possible area of the common part of
two triangles A'B'C' and KLM, if the points K,L and M
are lying on the segments [AB'], [CA'] and [BC'] respectively?
The 9-th competition -- Saratov, 1975.
form first day second day
8 205a 206 207 208a | 213 214 215
9 209 206 210 208b | 216 215 217
10 211 212 205b 208 | 214 218 219
205.
a)
The triangle ABC was turned around the centre of the outscribed
circle by the angle less than 180 degrees and thus was obtained the
triangle A1B1C1.
The corresponding segments [AB] and [A1B1]
intersect in the point C2;
[BC] and [B1C1] -- A2; [AC] and
[A1C1] -- B2.
Prove that the triangle A2B2C2 is similar to the triangle ABC.
b)
The quadrangle ABCD was turned around the centre of the outscribed
circle by the angle less than 180 degrees and thus was obtained
the quadrangle A1B1C1D1.
Prove that the points of intersection of the corresponding lines
( (AB) and (A1B1),
(BC) and (B1C1),
(CD) and (C1D1),
(DA) and (D1A1) )
are the vertices of the parallelogram.
206.
Given a triangle ABC with the unit area. The first player chooses a
point X on the side [AB], than the second -- Y on [BC] side, and,
finally, the first chooses a point Z on [AC] side. The first tries to
obtain the greatest possible area of the XYZ triangle, the second --
the smallest.
What area can obtain the first for sure and how?
207.
What is the smallest perimeter of the convex 32-angle, having all the
vertices in the nodes of cross-lined paper with the sides of its
squares equal to 1?
208.
a)
Given a big square consisting of 7x7 squares. You should mark the
centres of k points in such a way, that no quadruple of the marked
points will be the vertices of a rectangle with the sides parallel
to the sides of the given squares.
What is the greatest k such that the problem has solution?
b)
The same problem for 13x13 square.
209.
Denote the middles of the convex hexagon
A1A2A3A4A5A6 diagonals
A6A2, A1A3, A2A4, A3A5, A4A6, A5A1 as B1, B2, B3, B4,
B5, B6 respectively.
Prove that if the hexagon B1B2B3B4B5B6 is
convex, than its area equals to the quarter of the initial hexagon.
210.
Prove that it is possible to find 2n+1 of 2n digit numbers
containing only "1" and "2" as digits, such that every two of them
distinguish at least in 2n-1 digits.
211.
Given a finite set of polygons in the plane.
Every two of them have a common point.
Prove that there exists a straight line, that crosses all the polygons.
212.
Prove that for all the positive numbers a,b,c the following inequality
is valid:
a3+b3+c3+3abc>ab(a+b)+bc(b+c)+ac(a+c).
213.
Three flies are crawling along the perimeter of the ABC triangle in
such a way, that the centre of their masses is a constant point. One
of the flies has already passed along all the perimeter.
Prove that the centre of the flies' masses coincides with the centre of
masses of the ABC triangle. (The centre of masses for the triangle is
the point of medians intersection.)
214.
Several zeros, ones and twos are written on the blackboard. An
anonymous clean in turn pairs of different numbers, writing, instead of
cleaned, the number not equal to each. (0 instead of pair {1,2}; 1
instead of {0,2}; 2 instead of {0,1}).
Prove that if there remains one
number only, it does not depend on the processing order.
215.
Given a horizontal strip on the plane (its sides are parallel lines)
and n lines intersecting the strip. Every two of them intersect inside
the strip, and not a triple has a common point. Consider all the paths
along the segments of those lines, starting on the lower side of the
strip and ending on the upper side with the properties: moving along
such a path we are constantly rising up, and, having reached the
intersection, we are obliged to turn to another line.
Prove that:
a)
there are not less than n/2 such a paths without common points;
b)
there is a path consisting of not less than of n segments;
c)
there is a path that goes along not more than along n/2+1 lines;
d)
there is a path that goes along all the n lines.
216.
For what k is it possible to construct a cube kxkxk of the black
and white cubes 1x1x1 in such a way that every small cube has
the same colour, that have exactly two his neighbours. (Two cubes
are neighbours, if they have the common face.)
217.
Given a polynomial P(x) with
a)
natural coefficients;
b)
integer coefficients;
Let us denote with an the sum of the digits of P(n) value.
Prove that there is a number encountered in the sequence
a1, a2, ... , an, ... infinite times.
218.
The world and the european champion are determined in the same
tournament carried in one round. There are 20 teams and k of them are
european. The european champion is determined according to the results
of the games only between those k teams.
What is the greatest k such
that the situation, when the single european champion is the single
world outsider, is possible if:
a)
it is hockey (draws allowed)?
b)
it is volleyball (no draws)?
219.
a)
Given real numbers a1,a2,b1,b2 and positive p1,p2,q1,q2.
Prove that in the table 2x2
there is a number in the table, that is not less than another number
in the same row and is not greater than another number in the same
column (a saddle point).
b)
Given real numbers a1, a2, ... , an; b1, b2, ... , bn
and positive p1, p2, ... , pn; q1, q2, ... , qn.
We build the table nxn, with the numbers (0 < i,j <= n)
(ai + bj)/(pi + qj)
in the intersection of the i-th row and j-th column.
Prove that there is a number in the table, that is not less than
arbitrary number in the same row and is not greater than arbitrary
number in the same column (a saddle point).
The 10-th competition -- Dushanbe, 1976.
form first day second day
8 220 221 222ab 223 | 229 230 231
9 222b 224 223 225 | 230 232 231
10 223 226 227 228 225 | 233 234 231
220.
There are 50 exact watches lying on a table.
Prove that there exist a
certain moment, when the sum of the distances from the centre of the
table to the ends of the minute hands is more than the sum of the
distances from the centre of the table to the centres of the watches.
221.
A row of 1000 numbers is written on the blackboard. We write a new
row, below the first according to the rule: We write under every number
a the natural number, indicating how many times the number a is
encountered in the first line. Then we write down the third line:
under every number b -- the natural number, indicating how many times
the number b is encountered in the second line, and so on.
a)
Prove that there is a line that coincides with the preceding one.
b)
Prove that the eleventh line coincides with the twelfth.
c)
Give an example of the initial line such, that the tenth row differs
from the eleventh.
222.
Given three circumferences of the same radius in a plane.
a)
All three are crossing in one point K. Consider three arcs AK,CK,EK
: the A,C,E are the points of the circumferences intersection and
the arcs are taken in the clockwise direction. (Sorry, no picture.
Every arc is inside one circle, outside the second and on the border
of the third one)
Prove that the sum of the arcs is 180 degrees.
b)
Consider the case, when the three circles give a curvilinear
triangle BDF as there intersection (instead of one point K).
Prove that the sum of the AB, CD and EF arcs is 180 degrees.
(The arcs are taken in the clockwise direction. Every arc is inside
one circle, outside the second and on the border of the third one)
223.
The natural numbers x1 and x2 are less than 1000. We build a sequence:
Can You mark the cube's vertices with the three-digit binary
numbers in such a way, that the numbers at all the possible
couples of neighbouring vertices differ in at least two digits?
225.
Given 4 vectors a,b,c,d in the plane, such that a+b+c+d=0.
Prove the following inequality:
|a|+|b|+|c|+|d| >= |a+d|+|b+d|+|c+d|.
226.
Given right 1976-angle. The middles of all the sides and diagonals are
marked.
What is the greatest number of the marked points lying on one
circumference?
227.
There are n rectangles drawn on the rectangular sheet of paper with the
sides of the rectangles parallel to the sheet sides. The rectangles do
not have pairwise common interior points.
Prove that after cutting out
the rectangles the sheet will split into not more than n+1 part.
228.
There are three straight roads. Three pedestrians are moving
along those roads, and they are NOT on one line in the initial
moment.
Prove that they will be one line not more than twice.
229.
Given a chess-board 99x99 with a set F of fields marked on it (the set
is different in three tasks). There is a beetle sitting on every field
of the set F. Suddenly all the beetles have raised into the air and
flied to another fields of the same set. The beetles from the
neighbouring fields have landed either on the same field or on the
neighbouring ones (may be far from their starting point). (We consider
the fields to be neighbouring if they have at least one common vertex.)
Consider a statement:
"There is a beetle, that either stayed on the
same field or moved to the neighbouring one".
Is it always valid if
the figure F is:
a)
A central cross, i.e. the union of
the 50-th row and the 50-th column?
b)
A window frame, i.e. the union of
the 1-st, 50-th and 99-th rows and
the 1-st, 50-th and 99-th columns?
c)
All the chess-board?
230.
Let us call "big" a triangle with all sides longer than 1. Given a
equilateral triangle with all the sides equal to 5.
Prove that:
a)
You can cut 100 big triangles out of given one.
b)
You can divide the given triangle onto 100 big
nonintersecting ones fully covering the initial one.
c)
The same as b), but the triangles either do not have
common points, or have one common side, or one common vertex.
d)
The same as c), but the initial triangle has the side 3.
231.
Given natural n. We shall call "universal" such a sequence of natural
number a1, a2, ... , ak; k>=n, if we can obtain every transposition
of the first n natural numbers (i.e such a sequence of n numbers, that
every one is encountered only once) by deleting some its members.
(Examples: (1,2,3,1,2,1,3) is universal for n=3, and (1,2,3,2,1,3,1) --
not, because You can't obtain (3,1,2) from it.)
The goal is to estimate
the length of the shortest universal sequence for given n.
a)
Give an example of the universal sequence of n2 members.
b)
Give an example of the universal sequence of (n2 -n + 1) members.
c)
Prove that every universal sequence contains not less than
n(n + 1)/2 members
d)
Prove that the shortest universal sequence for n=4 contains
12 members
e)
Find as short universal sequence, as You can. The Organising
Committee knows the method for (n2 - 2n +4) members.
232.
n numbers are written down along the circumference. Their
sum equals to zero, and one of them equals 1.
a)
Prove that there are two neighbours with their difference
not less than n/4.
b)
Prove that there is a number that differs from the arithmetic
mean of its two neighbours not less than on 8/{n2}.
c)
Try to improve the previous estimation, i.e what number can be
used instead of 8?
d)
Prove that for n=30 there is a number that differs from the
arithmetic mean of its two neighbours not less than on 2/113;
give an example of such 30 numbers along the circumference,
that not a single number differs from the arithmetic mean of
its two neighbours more than on 2/113.
233.
Given right n-angle wit the point O -- its centre. All the vertices
are marked either with +1 or -1. We may change all the signs in the
vertices of right k-angle (2 <= k <= n) with the same centre O.
(By 2-angle we understand a segment, being halved by O.)
Prove that in a), b) and c) cases there exists such a set of
(+1)s and (-1)s, that we can never obtain a set of (+1)s only.
a)
n = 15;
b)
n = 30;
c)
n > 2;
d)
Let us denote K(n) the maximal number of (+1) and (-1) sets
such, that it is impossible to obtain one set from another.
Prove, for example, that K(200) = 280.
234.
Given a sphere of unit radius with the big circle (i.e of unit radius)
that will be called "equator". We shall use the words "pole",
"parallel","meridian" as self-explanatory.
a)
Let g(x), where x is a point on the sphere, be the distance from
this point to the equator plane.
Prove that g(x) has the property
if x1, x2, x3 are the ends of the
pairwise orthogonal radiuses, than
g(x1)2 + g(x2)2 + g(x3)2 = 1. (*)
Let function f(x) be an arbitrary nonnegative
function on a sphere that satisfies (*) property.
b)
Let x1 and x2 points be on the same meridian between the
north pole and equator, and x1 is closer to the pole than x2.
Prove that f(x1) > f(x2).
c)
Let y1 be closer to the pole than y2.
Prove that f(y1) > f(y2).
d)
Let z1 and z2 be on the same parallel.
Prove that f(z1) = f(z2).
e)
Prove that for all x , f(x) = g(x).
The 11-th competition -- Tallinn, 1977.
form first day second day
8 235 236 237b 238 | 243 244ab 245 246
9 237a 239 235 240 | 247 248 249 250
10 237a 239 241 242 235 | 251 244 246
235.
Given a closed broken line without self-intersections in a plane. Not
a triple of its vertices belongs to one straight line. Let us call
"special" a couple of line's segments if the one's continuation
intersects another.
Prove that there is even number of special pairs.
236.
Given several points, not all lying on one straight line. Some number
is assigned to every point. It is known, that if a straight line
contains two or more points, than the sum of the assigned to those
points equals zero.
Prove that all the numbers equal to zero.
237.
a)
Given a circle with two inscribed triangles T1 and T2. The
vertices of T1 are the middles of the arcs with the ends in the
vertices of T2. Consider a hexagon -- the intersection of T1 and
T2.
Prove that its main diagonals are parallel to T1 sides and
are intersecting in one point.
b)
The segment, that connects the middles of the arcs AB and AC of the
circle outscribed around the ABC triangle, intersects [AB] and [AC]
sides in D and K points.
Prove that the points A,D,K and O -- the
centre of the circle -- are the vertices of a diamond.
238.
Several black an white checkers (tokens?) are standing along the
circumference. Two men remove checkers in turn. The first removes
all the black ones that had at least one white neighbour, and the
second -- all the white ones that had at least one black neighbour.
They stop when all the checkers are of the same colour.
a)
Let there be 40 checkers initially. Is it possible that after
two moves of each man there will remain only one (checker)?
b)
Let there be 1000 checkers initially. What is the minimal possible
number of moves to reach the position when there will remain only
one (checker)?
239.
Given infinite sequence an. It is known that the limit of
bn=an+1-an/2 equals zero.
Prove that the limit of an equals zero.
240.
There are direct routes from every city of a certain country to every
other city. The prices are known in advance. Two tourists (they do not
necessary start from one city) have decided to visit all the cities,
using only direct travel lines. The first always chooses the cheapest
ticket to the city, he has never been before (if there are several --
he chooses arbitrary destination among the cheapests). The second --
the most expensive (they do not return to the first city).
Prove that
the first will spend not more money for the tickets, than the second.
/* The fact seems to be evident, but the proof is not easy -- VAP */
241.
Every vertex of a convex polyhedron belongs to three edges.
It is possible to outscribe a circle around all its faces.
Prove that the polyhedron can be inscribed in a sphere.
242.
The polynomial x10 + ?x9 + ?x8 + ... + ?x + 1 is written on the
blackboard. Two players substitute (real) numbers instead of one of the
question marks in turn. (9 turns total.) The first wins if the
polynomial will have no real roots.
Who wins?
243.
Seven elves are sitting at a round table. Each elf has a cup. Some
cups are filled with some milk. Each elf in turn and clockwise divides
all his milk between six other cups. After the seventh has done this,
every cup was containing the initial amount of milk.
How much milk did
every cup contain, if there was three litres of milk total?
244.
Let us call "fine" the 2n-digit number if it is exact square
itself and the two numbers represented by its first n digits
(first digit may not be zero) and last n digits (first digit may
be zero, but it may not be zero itself) are exact squares also.
a)
Find all two- and four-digit fine numbers.
b)
Is there any six-digit fine number?
c)
Prove that there exists 20-digit fine number.
d)
Prove that there exist at least ten 100-digit fine numbers.
e)
Prove that there exists 30-digit fine number.
245.
Given a set of n positive numbers. For each its nonempty subset
consider the sum of all the subset's numbers.
Prove that You can divide those sums onto n groups in such a way, that
the least sum in every group is not less than a half of the greatest
sum in the same group.
246.
There are 1000 tickets with the numbers 000, 001, ... , 999; and 100
boxes with the numbers 00, 01, ... , 99. You may put a ticket in a box,
if You can obtain the box number from the ticket number by deleting one
digit.
Prove that:
a)
You can put all the tickets in 50 boxes;
b)
40 boxes is not enough for that;
c)
it is impossible to use less than 50 boxes.
d)
Consider 10000 4-digit tickets, and You are allowed to delete two
digits. Prove that 34 boxes is enough for storing all the tickets.
e)
What is the minimal used boxes set in the case of k-digit tickets?
247.
Given a square 100x100 on the sheet of cross-lined paper. There are
several broken lines drawn inside the square. Their links consist of
the small squares sides. They are neither pairwise- nor
self-intersecting (have no common points). Their ends are on the big
square boarder, and all the other vertices are in the big square
interior.
Prove that there exists (in addition to four big square
angles) a node (corresponding to the cross-lining family, inside the
big square or on its side) that does not belong to any broken line.
248.
Given natural numbers x1,x2,...,xn;y1,y2,...,ym.
The following condition is valid:
(x1+x2+...+xn)=(y1+y2+...+ym)<mn. (*)
Prove that it is possible to delete some terms from (*) (not
all and at least one) and to obtain another valid condition.
249.
Given 1000 squares on the plane with their sides parallel to the
coordinate axes. Let M be the set of those squares centres.
Prove that You can mark some squares in such a way, that every point of
M will be contained not less than in one and not more than in four
marked squares.
250.
Given scales and a set of n different weights. We take weights in turn
and add them on one of the scales sides. Let us denote "L" the scales
state with the left side down, and "R" -- with the right side down.
a)
Prove that You can arrange the weights in such an order,
that we shall obtain the sequence LRLRLRLR... of the scales
states. (That means that the state of the scales will be
changed after putting every new weight.)
b)
Prove that for every n-letter word containing R's and L's
only You can arrange the weights in such an order, that the
sequence of the scales states will be described by that word.
251.
Let us consider one variable polynomials with the senior
coefficient equal to one. We shall say that two polynomials
P(x) and Q(x) commute, if P(Q(x))=Q(P(x)) (i.e. we obtain
the same polynomial, having collected the similar terms).
a)
For every a find all Q such that the Q degree is not
greater than three, and Q commutes with (x2 - a).
b)
Let P be a square polynomial, and k is a natural number. Prove that
there is not more than one commuting with P k-degree polynomial.
c)
Find the 4-degree and 8-degree polynomials
commuting with the given square polynomial P.
d)
R and Q commute with the same square polynomial P.
Prove that Q and R commute.
e)
Prove that there exists a sequence P2, P3, ... , Pn, ... (Pk is
k-degree polynomial), such that P2(x) = x2 - 2, and all the
polynomials in this infinite sequence pairwise commute.
The 12-th competition -- Tashkent, 1978.
form first day second day
8 252 253 254 255ab | 260 261 262 263
9 252 253 256 257 | 260 261 264 265
10 258 259 255cde257 | 260 266 267 268
252.
Let an be the closest to sqrt(n) integer.
Find the sum 1/a1 + 1/a2 + ... + 1/a1980.
253.
Given a quadrangle ABCD and a point M inside it such that ABMD is a
parallelogram. the angle CBM equals to CDM.
Prove that the angle ACD equals to BCM.
254.
Prove that there is no m such that (1978m - 1) is divisible
by (1000m - 1).
255.
Given a finite set K0 of points (in the plane or space). The sequence
of sets K1, K2, ... , Kn, ... is build according to the rule: we
take all the points of Ki, add all the symmetric points with respect
to all its points, and, thus obtain Ki+1.
a)
Let K0 consist of two points A and B with the distance 1 unit
between them. For what n the set Kn contains the point that is
1000 units far from A?
b)
Let K0 consist of three points that are the vertices of the
equilateral triangle with the unit square. Find the area of minimal
convex polygon containing Kn.
K0 below is the set of the unit volume tetrahedron vertices.
c)
How many faces contain the minimal convex polyhedron containing K1?
d)
What is the volume of the abovementioned polyhedron?
e)
What is the volume of the minimal convex polyhedron containing Kn?
256.
Given two heaps of checkers. the bigger contains m checkers, the
smaller -- n (m>n). Two players are taking checkers in turn from the
arbitrary heap. The players are allowed to take from the heap a number
of checkers (not zero) divisible by the number of checkers in another
heap. The player that takes the last checker in any heap wins.
a)
Prove that if m > 2n, than the first can always win.
b)
Find all x such that if m > xn, than the first can always win.
257.
Prove that there exists such an infinite sequence {xi}, that for
all m and all k (m<>k) holds the inequality |xm-xk|>1/|m-k|.
258.
Let f(x) = x2 + x + 1.
Prove that for every natural m>1 the numbers
m, f(m), f(f(m)), ... are relatively prime.
259.
Prove that there exists such a number A that You can inscribe 1978
different size squares in the plot of the function y = A sin(x). (The
square is inscribed if all its vertices belong to the plot.)
260.
Given three automates that deal with the cards with the pairs of
natural numbers. The first, having got the card with (a,b), produces
new card with (a+1,b+1); the second, having got the card with (a,b),
produces new card with (a/2,b/2), if both a and b are even and nothing
in the opposite case; the third, having got the pair of cards with
(a,b) and (b,c) produces new card with (a,c). All the automates return
the initial cards also. Suppose there was (5,19) card initially. Is it
possible to obtain
a)
(1,50)?
b)
(1,100)?
c)
Suppose there was (a,b) card initially (a<b). We want to obtain
(1,n) card. For what n is it possible?
261.
Given a circle with radius R and inscribed n-angle with area S.
We mark one point on every side of the given polygon.
Prove that the perimeter of the polygon with the vertices in the
marked points is not less than 2S/R.
262.
The checker is standing on the corner field of a nxn chess-board.
Each of two players moves it in turn to the neighbour (i.e. that
has the common side) field. It is forbidden to move to the field,
the checker has already visited. That who cannot make a move losts.
a)
Prove that for even n the first can always win, and if n is odd,
than the second can always win.
b)
Who wins if the checker stands initially on the neighbour to the
corner field?
263.
Given n nonintersecting segments in the plane. Not a pair of those
belong to the same straight line. We want to add several segments,
connecting the ends of given ones, to obtain one nonselfintersecting
broken line. Is it always possible?
264.
Given 0 < a <= x1 <= x2 <= ... <= xn <= b.
Prove that
Given a simple number p>3. Consider the set M of the pairs (x,y) with
the integer coordinates in the plane such that 0 <= x < p; 0 <= y < p.
Prove that it is possible to mark p points of M such that not a triple
of marked points will belong to one line and there will be no
parallelogram with the vertices in the marked points.
266.
Prove that for every tetrahedron there exist two planes such that the
projection areas on those planes relation is not less than sqrt(2).
267.
Given a1, a2, ... , an. Define bk = (a1 + a2 + ... + ak)/k
for 1 <= k <= n. Let
Consider a sequence xn=(1+sqrt(2)+sqrt(3))n.
Each member can be represented as
xn=qn+rnsqrt(2)+snsqrt(3)+tnsqrt(6),
where qn, rn, sn, tn are integers.
Find the limits of the fractions rn/qn,
sn/qn, tn/qn.
The 13-th competition -- Tbilisi, 1979.
form first day second day
8 269 270 271 | 274 275 276 277
9 269 272 271 | 278 279 280 281
10 273 272 271 | 276 275 282 283
269.
What is the least possible relation of two isosceles triangles areas,
if three vertices of the first one belong to three different sides of
the second one?
270.
A grasshopper is hopping in the angle x>=0, y>=0 of the coordinate
plane (that means that it cannot land in the point with negative
coordinate). If it is in the point (x,y), it can either jump to the
point (x+1,y-1), or to the point (x-5,y+7).
Draw a set of such an initial points (x,y), that having started from
there, a grasshopper cannot reach any point farther than 1000 from
the point (0,0). Find its area.
271.
Every member of a certain parliament has not more than 3 enemies.
Prove that it is possible to divide it onto two subparliaments so, that
everyone will have not more than one enemy in his subparliament. (A is
the enemy of B if and only if B is the enemy of A.)
272.
Some numbers are written in the notebook. We can add to that list the
arithmetic mean of some of them, if it doesn't equal to the number,
already having been included in it. Let us start with two numbers, 0
and 1.
Prove that it is possible to obtain
a)
1/5;
b)
an arbitrary rational number between 0 and 1.
273.
For every n the decreasing sequence {xk} satisfies a condition
x1+x4/2+x9/3+...+xn2/n<=1.
Prove that for every n it also satisfies
x1+x2/2+x3/3+...+xn/n<=3.
274.
Given some points in the plane. For some pairs A,B the vector AB is
chosen. For every point the number of the chosen vectors starting in
that point equal to the number of the chosen vectors ending in that
point.
Prove that the sum of the chosen vectors equals to zero vector.
275.
What is the least possible number of the checkers being required
a)
for the 8x8 chess-board;
b)
for the nxn chess-board;
to provide the property: Every line (of the chess-board fields)
parallel to the side or diagonal is occupied by at least one checker?
276.
Find x and y (a and b parameters):
(x-y*sqrt(x2-y2))/(sqrt(1-x2+y2)) = a;
(y-x*sqrt(x2-y2))/(sqrt(1-x2+y2)) = b.
277.
Given some square carpets with the total area 4.
Prove that they can fully cover the unit square.
278.
Prove that for the arbitrary numbers
x1, x2, ... , xn from the [0,1] segment
Natural p and q are relatively prime. The [0,1] is divided onto
(p+q) equal segments.
Prove that every segment except two marginal contain exactly one from
the (p+q-2) numbers {1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1q)}.
280.
Given the point O in the space and 1979 straight lines l1, l2, ... ,
l1979 containing it. Not a pair of lines is orthogonal.
Given a point A1 on l1 that doesn't coincide with O.
Prove that it is possible to choose the points Ai on li (i = 2, 3,
... , 1979) in so that 1979 pairs will be orthogonal:
A1A3 and l2;
A2A4 and l3;
..........
Ai-1Ai+1 and li;
..........
A1977A1979 and l1978;
A1978A1 and l1979;
A1979A2 and l1
281.
The finite sequence a1, a2, ... , an of ones and zeroes should
satisfy a condition:
for every k from 0 to (n-1) the sum
a1ak+1 + a2ak+2 + ... + an-kan
should be odd.
a)
Build such a sequence for n=25.
b)
Prove that there exists such a sequence for some n > 1000.
282.
The convex quadrangle is divided by its diagonals onto four
triangles. The circles inscribed in those triangles are equal.
Prove that the given quadrangle is a diamond.
283.
Given n points (in sequence) A1, A2, ... , An on a line. All the
segments A1A2, A2,A3, ... An-1An are shorter than 1. We need to
mark (k-1) points so that the difference of every two segments, with
the ends in the marked points, is shorter than 1.
Prove that it is possible
a)
for k=3;
b)
for every k less than (n-1).
The 14-th competition -- Saratov, 1980.
form first day second day
8 284 285 286 287 | 293 294 295 296
9 288 289 286 290 | 295 297 298 299
10 291 289 292 290 | 300 301 302 303
284.
All the two-digit numbers from 19 to 80 are written in a line without spaces.
Is the obtained number (192021....7980) divisible by 1980?
285.
The vertical side of a square is divided onto n segments. The sum of
the segments with even numbers lengths equals to the sum of the
segments with odd numbers lengths. (n-1) lines parallel to the
horizontal sides are drawn from the segments ends, and, thus, n strips
are obtained. The diagonal is drawn from the lower left corner to the
upper right one. This diagonal divides every strip onto left and right
parts.
Prove that the sum of the left parts of odd strips areas equals
to the sum of the right parts of even strips areas.
286.
The load for the space station "Salute" is packed in containers. There
are more than 35 containers, and the total weight is 18 metric tons.
There are 7 one-way transport spaceships "Progress", each able to bring
3 metric tons to the station. It is known that they are able to take an
arbitrary subset of 35 containers.
Prove that they are able to take all the load.
287.
The points M and P are the middles of [BC] and [CD] sides of a convex
quadrangle ABCD. It is known that |AM| + |AP| = a.
Prove that the ABCD area is less than {a2}/2.
288.
Are there three simple numbers x,y,z, such that x2 + y3 = z4?
289.
Given a point E on the diameter AC of the certain circle.
Draw a chord BD to maximise the area of the quadrangle ABCD.
290.
There are several settlements on the bank of the Big Round Lake. Some
of them are connected with the regular direct ship lines. Two
settlements are connected if and only if two next (counterclockwise) to
each ones are not connected.
Prove that You can move from the arbitrary settlement to another
arbitrary settlement, having used not more than three ships.
291.
The six-digit decimal number contains six different non-zero digits and
is divisible by 37.
Prove that having transposed its digits You can
obtain at least 23 more numbers divisible by 37.
292.
Find real solutions of the system
sin x + 2 sin(x+y+z) = 0,
sin y + 3 sin(x+y+z) = 0,
sin z + 4 sin(x+y+z) = 0.
293.
Given 1980 vectors in the plane, and there are some non-collinear among
them. The sum of every 1979 vectors is collinear to the vector not
included in that sum.
Prove that the sum of all vectors equals to the zero vector.
294.
Let us denote with S(n) the sum of all the digits of n.
a)
Is there such an n that n+S(n)=1980?
b)
Prove that at least one of two arbitrary successive natural
numbers is representable as n + S(n) for some third number n.
295.
Some squares of the infinite sheet of cross-lined paper are red.
Each 2x3 rectangle (of 6 squares) contains exactly two red squares.
How many red squares can be in the 9x11 rectangle of 99 squares?
296.
An epidemic influenza broke out in the elves city. First day some of
them were infected by the external source of infection and nobody later
was infected by the external source. The elf is infected when visiting
his ill friend. In spite of the situation every healthy elf visits all
his ill friends every day. The elf is ill one day exactly, and has the
immunity at least on the next day. There is no graftings in the city.
Prove that
a)
If there were some elves immunised by the external source on
the first day, the epidemic influenza can continue arbitrary
long time.
b)
If nobody had the immunity on the first day, the epidemic influenza
will stop some day.
297.
Let us denote with P(n) the product of all the digits of n. Consider
the sequence nk+1 = nk + P(nk).
Can it be unbounded for some n1?
298.
Given equilateral triangle ABC. Some line, parallel to [AC] crosses
[AB] and [BC] in M and P points respectively. Let D be the centre of
PMB triangle, E - the middle of the [AP] segment.
Find the angles of DEC triangle.
299.
Let the edges of rectangular parallelepiped be x,y and z (x<y<z).
Let p=4(x+y+z), s=2(xy+yz+zx) and
d=sqrt(x2+y2+z2) be its perimeter,
surface area and diagonal length, respectively.
Prove that
x < 1/3(p/4 - sqrt(d2 - s/2)),
z > 1/3(p/4 + sqrt(d2 - s/2)).
300.
The A set consists of integers only. Its minimal element is 1
and its maximal element is 100. Every element of A except 1 equals
to the sum of two (may be equal) numbers being contained in A.
What is the least possible number of elements in A?
301.
Prove that there is an infinite number of such numbers B that the
equation [x3/2] + [y3/2] = B has at least 1980 integer
solutions (x,y). ([z] denotes the greatest integer not exceeding z.)
302.
The edge [AC] of the tetrahedron ABCD is orthogonal to [BC], and [AD]
is orthogonal to [BD].
Prove that the cosine of the angle between (AC)
and (BD) lines is less than |CD|/|AB|.
303.
The number x from [0,1] is written as an infinite decimal fraction.
Having rearranged its first five digits after the point we can
obtain another fraction that corresponds to the number x1.
Having rearranged five digits of xk from (k+1)-th till (k+5)-th
after the point we obtain the number xk+1.
a)
Prove that the sequence xi has limit.
b)
Can this limit be irrational if we have started with the
rational number?
c)
Invent such a number, that always produces irrational numbers,
no matter what digits were transposed.
The 15-th competition -- Alma-Ata, 1981.
form first day second day
8 304 305 306 307 | 315 316 317 318
9 308 309 310 311 | 319 320 321 322
10 311 312 312 314 | 323 324 325 326
304.
Two equal chess-boards (8x8) have the same centre, but one is rotated
by 45 degrees with respect to another.
Find the total area of black
fields intersection, if the fields have unit length sides.
305.
Given points A,B,M,N on the circumference. Two chords [MA1] and [MA2]
are orthogonal to (NA) and (NB) lines respectively.
Prove that (AA1) and (BB1) lines are parallel.
306.
Let us say, that a natural number has the property P(k) if it can be
represented as a product of k succeeding natural numbers greater than 1.
a)
Find k such that there exists n which has properties
P(k) and P(k+2) simultaneously.
b)
Prove that there is no number having properties
P(2) and P(4) simultaneously.
307.
The rectangular table has four rows. The first one contains arbitrary
natural numbers (some of them may be equal). The consecutive lines are
filled according to the rule: we look through the previous row from
left to the certain number n and write the number k if n was met k
times.
Prove that the second row coincides with the fourth one.
308.
Given real a. Find the least possible area of the rectangle
with the sides parallel to the coordinate axes and containing
the figure determined by the system of inequalities
y <= -x2,
y >= x2 - 2x + a.
/* as there is no derivatives in the school program of the 9-th
form, the participants had to use parabola properties -VAP */
309.
Three equilateral triangles ABC, CDE, EHK (the vertices are mentioned
counterclockwise) are lying in the plane so, that the vectors [AD[ and
[DK[ are equal.
Prove that the triangle BHD is also equilateral.
310.
There are 1000 inhabitants in a settlement. Every evening every
inhabitant tells all his friends all the news he had heard the previous
day. Every news becomes finally known to every inhabitant.
Prove that
it is possible to choose 90 of inhabitants so, that if You tell them a
news simultaneously, it will be known to everybody in 10 days.
311.
It is known about real a and b that the inequality
a cos(x) + b cos(3x) > 1 has no real solutions.
Prove that |b| <=1.
312.
The points K and M are the centres of the AB and CD sides of the convex
quadrangle ABCD. The points L and M belong to two other sides and KLMN
is a rectangle.
Prove that KLMN area is a half of ABCD area.
313.
Find all the sequences of natural kn with two properties:
for all n kn <= n sqrt(n);
for all m>n (kn - km) is divisible by (m-n).
314.
Is it possible to fill a rectangular table with black and white
squares (only) so, that the number of black squares will equal
to the number of white squares, and each row and each column
will have more than 75% squares of the same colour?
315.
The quadrangles AMBE, AHBT, BKXM, and CKXP are parallelograms.
Prove that the quadrangle ABTE is also parallelogram.
(the vertices are mentioned counterclockwise)
316.
Find the natural solutions of the equation x3 - y3 = xy + 61.
317.
Eighteen soccer teams have played 8 tours of a one-round tournament.
Prove that there is a triple of teams, having not met each other yet.
318.
The points C1, A1, B1 belong to [AB], [BC], [CA] sides, respectively,
of the ABC triangle.
|AC1|/|C1B| =
|BA1|/|A1C| =
|CB1|/|B1A| = 1/3.
Prove that the perimeter P of the ABC triangle and the perimeter p
of the A1B1C1 triangle satisfy inequality
P/2 < p < 3P/4.
319.
Positive numbers x,y satisfy equality x3+y3=x-y.
Prove that x2+y2<1.
320.
A pupil has tried to make a copy of a convex polygon, drawn inside
the unit circle. He draw one side, from its end -- another, and so on.
Having finished, he has noticed that the first and the last vertices
do not coincide, but are situated d units of length far from each other.
The pupil draw angles prec
