Navvye/Olympiad-Problems-CSV huggingface dataset:
1986 Tournament Of Towns
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that

(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.

(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.



( V . V . Proizvolov , Moscow)
1986 Tournament Of Towns
A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins?



{S. Fomin , Leningrad)
1986 Tournament Of Towns
Consider subsets of the set $1 , 2,..., N$.

For each such subset we can compute the product of the reciprocals of each member.

Find the sum of all such products.
1986 Tournament Of Towns
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .



(A. Andjans, Riga)
1986 Tournament Of Towns
In a football tournament of one round (each team plays each other once, $2$ points for win , $1$ point for draw and $0$ points for loss), $28$ teams compete. During the tournament more than $75\%$ of the matches finished in a draw . Prove that there were two teams who finished with the same number of points.



(M . Vora, gymnasium student , Hungary)
1986 Tournament Of Towns
Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance.



(A . K . Tolpugo , Kiev)
1986 Tournament Of Towns
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
1986 Tournament Of Towns
Does there exist a number $N$ so that there are $N - 1$ infinite arithmetic progressions with differences $2 , 3 , 4 ,..., N$ , and every natural number belongs to at least one of these progressions?
1986 Tournament Of Towns
Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?
1986 Tournament Of Towns
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even .

For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot  5\cdot 3 \cdot 1$ .

Prove that $1986 !! + 1985 !!$ i s divisible by $1987$.



(V.V . Proizvolov , Moscow)
1986 Tournament Of Towns
Squares of an $8 \times 8$ chessboard are each allocated a number between $1$ and $32$ , with each number being used twice. Prove that it is possible to choose $32$ such squares, each allocated a different number, so that there is at least one such square on each row or column .



(A . Andjans, Riga
1986 Tournament Of Towns
On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ .



( A . F . Sidorenko)
1987 Tournament Of Towns
Prove that for all values of $a$, $3(1+a^2+a^4) \ge (1+a+a^2)^2$ .
1987 Tournament Of Towns
In an acute angled triangle the bases of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .
1987 Tournament Of Towns
We are given two three-litre bottles, one containing $1$ litre of water and the other containing $1$ litre of $2\%$ salt solution . One can pour liquids from one bottle to the other and then mix them to obtain solutions of different concentration . Can one obtain a $1 . 5\%$ solution of salt in the bottle which originally contained water?



(S . Fomin, Leningrad),
1987 Tournament Of Towns
We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles?



( S . Fomin , Leningrad)
1987 Tournament Of Towns
A machine gives out five pennies for each nickel inserted into it and five nickels for each penny. Can Peter , who starts out with one penny, use the machine several times in such a way as to end up with an equal number of nickels and pennies?



(F. Nazarov, Leningrad Olympiad, 1987)
1987 Tournament Of Towns
Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals

(a)  four

(b ) five possess an inscribed circle?



(P. M . Sedrakyan , Yerevan)
1987 Tournament Of Towns
Nine pawns forming a $3$ by $3$ square are placed in the lower left hand corner of an $8$ by $8$ chessboard. Any pawn may jump over another one standing next to it into a free square, i .e. may be reflected symmetrically with respect to a neighb our's centre (jumps may be horizontal , vertical or diagonal) . It is required to rearrange the nine pawns in another corner of the chessboard (in another $3$ by $3$ square) by means of such jumps. Can the pawns be thus re-arranged in the

(a) upper left hand corner?

(b) upper right hand corner?



(J . E . Briskin)
1987 Tournament Of Towns
Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre.



(V . Pogrebnyak , year 12 student , Vinnitsa,)
1987 Tournament Of Towns
A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the

top faces of all the cubes are the same colour) by means of such operations?



( D. Fomin , Leningrad)
1987 Tournament Of Towns
Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?
1987 Tournament Of Towns
In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .
1987 Tournament Of Towns
On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$.



(A. Andj ans, Riga)
1987 Tournament Of Towns
Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ .



(Folklore)
1987 Tournament Of Towns
Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .
1987 Tournament Of Towns
In a certain city only simple (pairwise) exchanges of apartments are allowed (if two families exchange fiats , they are not allowed to participate in another exchange on the same day). Prove that any compound exchange may be effected in two days. It is assumed that under any exchange (simple or comp ound) each family occupies one fiat before and after the exchange and the family cannot split up .



(A . Shnirelman , N .N . Konstantinov)
1987 Tournament Of Towns
For any natural $n$ prove the inequality

$$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$
1987 Tournament Of Towns
Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle .



(J . Tabov , Sofia)
1987 Tournament Of Towns
Two players play a game on an $8$ by $8$ chessboard according to the following rules. The first player places a knight on the board. Then each player in turn moves the knight , but cannot place it on a square where it has been before. The player who has no move loses. Who wins in an errorless game , the first player or the second one? (The knight moves are the normal ones. )



(V . Zudilin , year 12 student , Beltsy (Moldova))
1987 Tournament Of Towns
Prove that the second last digit of each power of three is even .



(V . I . Plachkos)
1987 Tournament Of Towns
Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .
1987 Tournament Of Towns
In a game two players alternately choose larger natural numbers. At each turn the difference between the new and the old number must be greater than zero but smaller than the old number. The original number is 2. The winner is considered to be the player who chooses the number $1987$. In a perfect game, which player wins?
1987 Tournament Of Towns
We are given a figure bounded by arc $AC$ of a circle, and a broken line $ABC$, with the arc and broken line being on opposite sides of the chord $AC$. Construct a line passing through the mid-point of arc $AC$ and dividing the area of the figure into two regions of equal area.
1987 Tournament Of Towns
We are given three non-negative numbers $A , B$ and $C$ about which it is known that $A^4 + B^4 + C^4 \le  2(A^2B^2 + B^2C^2 + C^2A^2)$

(a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others.

(b) Prove that $A^2 + B^2 + C^2 \le  2(AB + BC + CA)$ .

(c) Does the original inequality follow from the one in (b)?



(V.A. Senderov , Moscow)
1987 Tournament Of Towns
There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ .



(A. Razborov)
1987 Tournament Of Towns
Three triangles (blue, green and red) have a common interior point $M$. Prove that it is possible to choose one vertex from each triangle so that point $M$ is located inside the triangle formed by these selected vertices.



(Imre Barani, Hungary)
1987 Tournament Of Towns
From vertex $A$ in square $ABCD$  (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four bases of these perpendiculars.
1987 Tournament Of Towns
In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?
1987 Tournament Of Towns
Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .
1987 Tournament Of Towns
From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the base of the perpendicular dropped to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.
1987 Tournament Of Towns
Consider the set of all pairs of positive integers $(A , B)$ in which $A < B$ . Some of these pairs are to $be$ designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form $A, A + D, A + 2D$, in which $D > 0$, the associated pairs $(A, A + D )$ , $(A , A + 2D)$ and $(A + D, A + 2D)$ would include at least one pair of each colour?
1987 Tournament Of Towns
An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?
1987 Tournament Of Towns
A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim?



(A. Andjans, Riga)
1988 Tournament Of Towns
In January Kolya and Vasya have been assessed at school $20$ times and each has been given $20$ marks (each being an integer no greater than $5$ , with both Kolya and Vasya receiving at least twos on each occasion). Kolya has been given as many fives as Vasya fours, as many fours as Vasya threes, as many threes as Vasya twos and as many twos as Vasya fives. If each has the same average mark , determine how many twos were given to Kolya.



(S . Fomin, Leningrad)
1988 Tournament Of Towns
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .
1988 Tournament Of Towns
(a) The vertices of a regular $10$-gon are painted in turn black and white. Two people play the following game . Each in turn draws a diagonal connecting two vertices of the same colour . These diagonals must not intersect . The winner is the player who is able to make the last move. Who will win if both players adopt the best strategy?

(b) Answer the same question for the regular $12$-gon .



(V.G. Ivanov)
1988 Tournament Of Towns
The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.
1988 Tournament Of Towns
We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number.



(Folklore)
1988 Tournament Of Towns
We are given triangle $ABC$. Two lines, symmetric with $AC$, relative to lines $AB$ and $BC$ are drawn, and meet at $K$ . Prove that the line $BK$ passes through the centre of the circumscribed circle of triangle $ABC$.



(V.Y. Protasov)
1988 Tournament Of Towns
Find all real solutions of the system of equations

$(x_3 + x_4 + x_5)^5 = 3x_1$

$(x_4 + x_5 + x_1)^5 = 3x_2$

$(x_5 + x _1 + x_2)^5 = 3x_3$

$(x_1 + x_2 + x_3)^5 = 3x_4$

$(x_2 + x_3 + x_4)^5 = 3x_5$



(L. Tumescu , Romania)
1988 Tournament Of Towns
We have a set of weights with masses $1$ gm, $2$ gm, $4$ gm and so on, all values being powers of $2$ . Some of these weights may have equal mass. Some weights were put on both sides of a balance beam, resulting in equilibrium. It is known that on the left hand side all weights were distinct . Prove that on the right hand side there were no fewer weights than on the left hand side.
1988 Tournament Of Towns
Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point?



( N . Vasiliev)
1988 Tournament Of Towns
The first quadrant of the Cartesian $0-x-y$ plane can be considered to be divided into an infinite set of squares of unit side length, arranged in rows and columns , formed by the axes and lines $x = i$ and $y = j$ , where $i$ and $j$ are non-negative integers. Is it possible to write a natural number $(1,2, 3,...)$ in each square , so that each row and column contains each natural number exactly once?



(V . S . Shevelev)
1988 Tournament Of Towns
Consider a sequence of words each consisting of two letters, $A$ and $B$ . The first word is "$A$" , while the second word is "$B$" . The $k$-th word is obtained from the ($k -2$)-nd by writing after it the ($k -1$)th one. (So the first few elements of the sequence are "$A$" , "$B$" ,"$AB$" , "$BAB$" , "$ABBAB$" . ) Does there exist in this sequence a "periodical" word, i.e. a word of the form $P P P ... P$ , where $P$ is a word , repeated at least once?



(Remark: For instance, the word $BABBBABB$ is of the form $PP$ , in which $P$ is repeated exactly once . )



(A. Andjans, Riga)
1988 Tournament Of Towns
Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?
1988 Tournament Of Towns
Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.
1988 Tournament Of Towns
The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements?



(S. Fomin , Leningrad)
1988 Tournament Of Towns
Pawns are placed on an infinite chess board so that they form an infinite square net (along any row or column containing pawns ther is a pawn , three free squares , pawn , three squares, and so on , with only every fourth row and every fourth column containing pawns). Prove that it is not possible for a knight to tour every free square once and only once.



(An old problem of A . K . Tolpugo)
1988 Tournament Of Towns
Determine the ratio of the bases (parallel sides) of the trapezoid for which there exists a line with $6$ points of intersection with the diagonals, lateral sides and extended bases cut $5$ equal segments?



( E . G . Gotman)
1988 Tournament Of Towns
It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .
1988 Tournament Of Towns
There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)
1988 Tournament Of Towns
A $20 \times 20  \times 20$ cube is composed of $2000$ bricks of size $2  \times 2  \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick .



(A . Andjans , Riga)
1988 Tournament Of Towns
Consider a sequence of words , consisting of the letters $A$ and $B$ .

The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$

(a) In which place of this sequence is the $1000$-th letter $A$?

(b ) Prove that this sequence is not periodic.



(V . Galperin , Moscows)
1988 Tournament Of Towns
It is known that the proportion of people with fair hair among people with blue eyes is more than the proportion of people with fair hair among all people. Which is greater , the proportion of people with blue eyes among people with fair hair, or the proportion of people with blue eyes among all people?



(Folklore)
1988 Tournament Of Towns
In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle.
1988 Tournament Of Towns
Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.
1988 Tournament Of Towns
Each face of a cube has been divided into four equal quarters and each quarter is painted with one of three available colours. Quarters with common sides are painted with different colours . Prove that each of the available colours was used in painting $8$ quarters.
1988 Tournament Of Towns
One of the numbers $1$ or $-1$ is assigned to each vertex of a cube. To each face of the cube is assigned the integer which is the product of the four integers at the vertices of the face. Is it possible that the sum of the $14$ assigned integers is $0$?



(G. Galperin)
1988 Tournament Of Towns
A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.
1988 Tournament Of Towns
Let $a_1 , a_2 ,... , a_n$ be an arrangement of the integers $1,2,..., n$. Let $$S=\frac{a_1}{1}+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{1}.$$Find a natural number $n$ such that among the values of $S$ for all arrangements $a_1 , a_2 ,... , a_n$ , all the integers from $n$ to $n + 100$ appear .
1988 Tournament Of Towns
(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel?

(The cogwheels may be rotated about their common axis, but not turned over.)



(b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.
1988 Tournament Of Towns
A convex $n$-vertex polygon is partitioned into triangles by nonintersecting diagonals . The following operation, called perestroyka (=reconstruction) , is allowed: two triangles $ABD$ and $BCD$ with a common side may be replaced by the triangles $ABC$ and $ACD$. By $P(n)$ denote the smallest number of perestroykas needed to transform any partitioning into any other one. Prove that

(a) $P ( n ) \ge  n - 3$

(b) $P (n) \le 2n - 7$

(c) $P(n) \le 2n - 10$ if $n \ge 13$ .



( D.Fomin , based on ideas of W. Thurston , D . Sleator, R. Tarjan)
1988 Tournament Of Towns
Does there exist a natural number which is not a divisor of any natural number whose decimal expression consists of zeros and ones, with no more than $1988$ ones?
1988 Tournament Of Towns
Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?
1988 Tournament Of Towns
Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.