Navvye/Olympiad-Problems-CSV huggingface dataset:
1988 Tournament Of Towns
Prove that for each vertex of a polyhedron it is possible to attach a natural number so that for each pair of vertices with a common edge, the attached numbers are not relatively prime (i.e. they have common divisors), and with each pair of vertices without a common edge the attached numbers are relatively prime.



(Note: there are infinitely many prime numbers.)
1988 Tournament Of Towns
A page of an exercise book is painted with $23$ colours, arranged in squares. A pair of colours is called good  if there are neighbouring squares painted with these colours. What is the minimum number of good pairs?
1988 Tournament Of Towns
What is the smallest number of squares of a chess board that can be marked in such a manner that

(a) no two marked squares may have a common side or a common vertex, and

(b) any unmarked square has a common side or a common vertex with at least one marked square?

Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice.



(A. Andjans, Riga)
1988 Tournament Of Towns
Prove that $a^2pq + b^2qr + c^2rp \le 0$, whenever $a, b$ and $c$ are the lengths of the sides of a triangle and $p + q + r = 0$ .



( J. Mustafaev , year 12 student, Baku)
1988 Tournament Of Towns
The integers $1 , 2,..., n$ are rearranged in such a way that if the integer $k, 1 \le k\le  n$, is not the first term, then one of the integers $k + 1$ or $k-1$ occurs to the left of $k$ . How many arrangements of the integers $1 , 2,..., n$ satisfy this condition?



(A. Andjans, Riga)
1988 Tournament Of Towns
There are $1988$ towns and $4000$ roads in a certain country (each road connects two towns) . Prove that there is a closed path passing through no more than $20$ towns.



(A. Razborov , Moscow)
1988 Tournament Of Towns
$M$ is an interior point of a rectangle $ABCD$ and $S$ is its area.

Prove that $S \le AM \cdot CM + BM \cdot  DM$.



(I.J . Goldsheyd)
1990 Tournament Of Towns
For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left(  \frac{1}{n-1} + \frac12 \right)^2+ \left(  \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$(S. Manukian, Yerevan)
1990 Tournament Of Towns
Two circles $c$ and $d$ are situated in the plane each outside the other. The points $C$ and $D$ are located on circles $c$ and $d$ respectively, so as to be as far apart as possible. Two smaller circles are constructed inside $c$ and $d$. Of these the first circle touches $c$ and the two tangents drawn from $C$ to $d$, while the second circle touches $d$ and the two tangents from $D$ to $c$. Prove that the small circles are equal.



(J. Tabov, Sofia)
1990 Tournament Of Towns
Is it possible to put together $27$ equal cubes, $9$ red, $9$ blue and $9$ white, so as to obtain a big cube in which each row (parallel to an arbitrary edge of the cube) contains three cubes with exactly two different colours?



(S. Fomin, Leningrad)
1990 Tournament Of Towns
A set of $61$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $59$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.)



(D. Fomin, Leningrad)
1990 Tournament Of Towns
Find the maximum number of parts into which the $Oxy$-plane can be divided by $100$ graphs of different quadratic functions of the form $y = ax^2 + bx + c$.



(N.B. Vasiliev, Moscow)
1990 Tournament Of Towns
If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one.



(A. Savin, Moscow)
1990 Tournament Of Towns
Fifteen elephants stand in a row. Their weights are expressed by integer numbers of kilograms. The sum of the weight of each elephant (except the one on the extreme right) and the doubled weight of its right neighbour is exactly $15$ tonnes. Determine the weight of each elephant.



(F.L. Nazarov)
1990 Tournament Of Towns
Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line.



(D. Fomin, Leningrad)
1990 Tournament Of Towns
Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$



(recalling that $ \sqrt2 = 1.414213..$.).



(D. Fomin, Leningrad)
1990 Tournament Of Towns
We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.)

(a)	Find an example of a basic collection other than the collection of $200$ weights each of value $1$.

(b)	How many different basic collections are there?



(D. Fomin, Leningrad)
1998 Tournament Of Towns
Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words : Anya listed the most, Vasya the least . They were awarded points as follows. Each word listed by only one of them scored $2$ points for this child. Each word listed by two of them scored $1$ point for each of these two children. Words listed by all three of them scored $0$ points. Is it possible that Vasya got the highest score, and Anya

the lowest?



(A Shapovalov)
1998 Tournament Of Towns
A chess king tours an entire $8\times 8$ chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves.



(V Proizvolov)
1998 Tournament Of Towns
$AB$ and $CD$ are segments lying on the two sides of an angle whose vertex is $O$. $A$ is between $O$ and $B$, and $C$ is between $O$ and $D$ . The line connecting the midpoints of the segments $AD$ and $BC$ intersects $AB$ at $M$ and $CD$ at $N$. Prove that $\frac{OM}{ON}=\frac{AB}{CD}$



(V Senderov)
1998 Tournament Of Towns
For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result?



(G Galperin)
1998 Tournament Of Towns
Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying?



(A Shapovalov)
1998 Tournament Of Towns
Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers?



(Folklore)
1998 Tournament Of Towns
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle  AMO$ where $O$ is the point of intersection of the diagonals of the parallelogram. Prove that $MD = MG$.



(M Smurov)
1998 Tournament Of Towns
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.)



(G Galperin)
1998 Tournament Of Towns
A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction?



(B, Frenkin)
1998 Tournament Of Towns
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?



(I Rubanov)
1998 Tournament Of Towns
$10$ people are sitting at a round table. There are some nuts in front of each of them, $100$ nuts altogether. After a certain signal each person passes some of his nuts to the person sitting to his right . If he has an even number of nuts, he passes half of them; otherwise he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone will have exactly $10$ nuts.



(A Shapovalov)
1998 Tournament Of Towns
Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying?



(A Fedotov)
1998 Tournament Of Towns
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result?



( G Galperin)
1998 Tournament Of Towns
What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?



(A Shapovalov)
1998 Tournament Of Towns
For some positive numbers $A, B, C$ and $D$, the system of equations



$x^2 + y^2 = A$ and $|x| + |y| = B$



has $m$ solutions,  while the system of equations



$x^2 + y^2 +z^2= X$ and $|x| + |y| +|z| = D$



has $n$ solutions. If $m > n > 1$, find $m$ and $n$.



( G Galperin)
1998 Tournament Of Towns
A circle with center $O$ is inscribed in an angle. Let $A$ be the reflection of $O$ across one side of the angle. Tangents to the circle from $A$ intersect the other side of the angle at points $B$ and $C$. Prove that the circumcenter of triangle $ABC$ lies on the bisector of the original angle.



(I.Sharygin)
1998 Tournament Of Towns
Prove that $$\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\geq \frac{a+b+c}{3}$$for positive reals $a,b,c$



(S Tokarev)
1998 Tournament Of Towns
A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ .



(Folklore)
1998 Tournament Of Towns
(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard.

We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$?



(b) The numbers $1 , 2, 22, 23 , . . . , 210$ are written on a blackboard.

We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have?



(A.Shapovalov)
1998 Tournament Of Towns
A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$ BNC$ and $DNA$ are equilateral.



(I.Sharygin)
1998 Tournament Of Towns
A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths?



(A Shapovalov)
1998 Tournament Of Towns
(a) Two people perform a card trick. The first performer takes $5$ cards from a $52$-card deck (previously shuffled by a member of the audience) , looks at them, and arranges them in a row from left to right: one face down (not necessarily the first one) , the others face up . The second performer guesses correctly the card which is face down. Prove that the performers can agree on a system which always makes this possible.

(b) For their second trick, the first performer arranges four cards in a row, face up, the fifth card is kept hidden. Can they still agree on a system which enables the second performer to correctly guess the hidden card?



(G Galperin)
1998 Tournament Of Towns
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.
1998 Tournament Of Towns
The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.
1998 Tournament Of Towns
In a triangle $ ABC$ the points $ A'$, $ B'$ and $ C'$ lie on the sides $ BC$, $ CA$ and $ AB$, respectively. It is known that $ \angle AC'B' = \angle B'A'C$, $ \angle CB'A' = \angle A'C'B$ and $ \angle BA'C' = \angle C'B'A$. Prove that $ A'$, $ B'$ and $ C'$ are the midpoints of the corresponding sides.
1998 Tournament Of Towns
Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?
1998 Tournament Of Towns
Let $ n$ and $ m$ be given positive integers. In one move, a chess piece called an $ (n,m)$-crocodile goes $ n$ squares horizontally or vertically and then goes $ m$ squares in a perpendicular direction. Prove that the squares of an infinite chessboard can be painted in black and white so that this chess piece always moves from a black square to a white one or vice-versa.
1998 Tournament Of Towns
John and Mary each have a white $8 \times  8$ square divided into $1 \times  1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells.



(A Shapovalov)
1998 Tournament Of Towns
Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral.



(P Kozhevnikov)
1998 Tournament Of Towns
All the diagonals of a regular $25$-gon are drawn. Prove that no $9$ of the diagonals pass through one interior point of the $25$-gon.



(A Shapovalov)
1998 Tournament Of Towns
There are $20$ beads of $10$ colours, two of each colour. They are put in $10$ boxes. It is known that one bead can be selected from each of the boxes so that each colour is represented. Prove that the number of such selections is a non-zero power of $2$.



(A Grishin)
1998 Tournament Of Towns
A gang of robbers took away a bag of coins from a merchant . Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies) . Prove that after one coin is removed, the number of the remaining coins is divisible by the number of robbers.



(Folklore, modified by A Shapovalov)
1998 Tournament Of Towns
Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, . . . , $19$ gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight .



(V Proizvolov)
1998 Tournament Of Towns
On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.



(P Kozhevnikov)
1998 Tournament Of Towns
On an $8 \times  8$ chessboard, $17$ cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other.



(R Zhenodarov)
1998 Tournament Of Towns
Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$ , find the one for which $x_1 x_2... x{20}$ is maximal.



(A Cherniatiev)
1998 Tournament Of Towns
The intelligence quotient (IQ) of a country is defined as the average IQ of its entire population. It is assumed that the total population and individual IQs remain constant throughout.

(a) (i) A group of people from country $A$ has emigrated to country $B$ . Show that it can happen that as a result , the IQs of both countries have increased.

(ii) After this, a group of people from $B$, which may include immigrants from $A$, emigrates to $A$. Can it happen that the IQs of both countries will increase again?

(b) A group of people from country $A$ has emigrated to country $B$, and a group of people from $B$ has emigrated to country $C$ . It is known that a s a result , the IQs o f all three countries have increased. After this, a group of people from $C$ emigrates to $B$ and a group of people from $B$ emigrates to $A$. Can it happen that the IQs of all three countries will increase again?



(A Kanel, B Begun)
1998 Tournament Of Towns
(a) Prove that for any two positive integers a and b the equation  $lcm (a, a + 5) =  lcm (b, b + 5)$ implies $a = b$.

(b) Is it possible that  $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$?



(A Shapovalov)



PS. part (a) for Juniors, both part for Seniors
1998 Tournament Of Towns
Nine numbers are arranged in a square table:

$a_1 \,\,\, a_2 \,\,\,a_3$

$b_1 \,\,\,b_2 \,\,\,b_3$

$c_1\,\,\, c_2 \,\,\,c_3$ .

It is known that the six numbers obtained by summing the rows and columns of the table are equal:

$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .

Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:

$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .



(V Proizvolov)
1998 Tournament Of Towns
Twelve places have been arranged at a round table for members of the Jury, with a name tag at each place . Professor K. being absent-minded instead of occupying his place, sits down at the next place (clockwise) . Each of the other Jury members in turn either occupies the place assigned to this member or, if it has been already occupied, sits down at the first free place in the clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements of this kind are possible?



(A Shapovalov)
1998 Tournament Of Towns
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?



(A Shen)
1998 Tournament Of Towns
In a function $f (x) = (x_2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:

(i) there is a numerical interval without any values of $f(x)$ ,

(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .



(A Kanel)
1999 Tournament Of Towns
A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?



(Tairova)
1999 Tournament Of Towns
$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$.



(A Blinkov)
1999 Tournament Of Towns
Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards.



(H. Lebesgue - A Kanel)
1999 Tournament Of Towns
A black unit equilateral triangle is drawn on the plane. How can we place nine tiles, each a unit equilateral triangle, on the plane so that they do not overlap, and each tile covers at least one interior point of the black triangle?



(Folklore)
1999 Tournament Of Towns
A square is cut into $100$ rectangles by $9$ straight lines parallel to one of the sides and $9$ lines parallel to another. If exactly $9$ of the rectangles are actually squares, prove that at least two of these $9$ squares are of the same size .



(V Proizvolov)
1999 Tournament Of Towns
There is  $500$ dollars in a bank. Two bank operations are allowed: to withdraw  $300$ dollars  from the bank or to deposit  $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done?



(AK Tolpygo)
1999 Tournament Of Towns
Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$.



(A Zaslavskiy)
1999 Tournament Of Towns
Two players play the following game. The first player starts by writing either $0$ or $1$ and then, on his every move, chooses either $0$ or $1$ and writes it to the right of the existing digits until there are $1999$ digits. Each time the first player puts down a digit (except the first one) , the second player chooses two digits among those already written and swaps them. Can the second player guarantee that after his last move the line of digits will be symmetrical about the middle digit?



(I Izmestiev)