Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Tournament Of Towns
There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end?



(3 points)
2007 Tournament Of Towns
Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible).



(4 points)
2008 Tournament Of Towns
Each of ten boxes contains a different number of pencils. No two pencils in the same box are of the same colour. Prove that one can choose one pencil from each box so that no two are of the same colour.
2008 Tournament Of Towns
Twenty-five of the numbers $1, 2, \cdots , 50$ are chosen. Twenty-five of the numbers$ 51, 52, \cdots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers.
2008 Tournament Of Towns
Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$
2008 Tournament Of Towns
Given three distinct positive integers such that one of them is the average of the two others. Can the product of these three integers be the perfect 2008th power of a positive integer?
2008 Tournament Of Towns
On a straight track are several runners, each running at a different constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.
2008 Tournament Of Towns
$100$ Queens are placed on a $100 \times 100$ chessboard so that no two attack each other. Prove that each of four $50 \times 50$ corners of the board contains at least one Queen.
2008 Tournament Of Towns
Each of $4$ stones weights the integer number of grams. A balance with arrow indicates the difference of weights on the left and the right sides of it. Is it possible to determine the weights of all stones in $4$ weighings, if the balance can make a mistake in $1$ gram in at most one weighing?
2008 Tournament Of Towns
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).
2008 Tournament Of Towns
Baron Munchausen claims that he got a map of a country that consists of five cities. Each two cities are connected by a direct road. Each road intersects no more than one another road (and no more than once). On the map, the roads are colored in yellow or red, and while circling any city (along its border) one can notice that the colors of crossed roads alternate. Can Baron's claim be true?
2008 Tournament Of Towns
Let $a_1,a_2,\cdots,a_n$ be a sequence of positive numbers, so that $a_1 + a_2 +\cdots + a_n \leq \frac 12$. Prove that

$$(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.$$



RemarkRemark. I think this problem was posted before, but I can't find the link now.
2008 Tournament Of Towns
Let $ABC$ be a non-isosceles triangle. Two isosceles triangles $AB'C$ with base $AC$ and $CA'B$ with base $BC$ are constructed outside of triangle $ABC$. Both triangles have the same base angle $\varphi$. Let $C_1$ be a point of intersection of the perpendicular from $C$ to $A'B'$ and the perpendicular bisector of the segment $AB$. Determine the value of $\angle AC_1B.$
2008 Tournament Of Towns
In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows:



- if the greatest odd divisor of

$n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$



- and if this residue equals $3$, then $a_n = a_{n-1} - 1.$





Prove that in this sequence



(a)  the number $1$ occurs infinitely many times;



(b) each positive integer occurs infinitely many times.



(The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )
2008 Tournament Of Towns
Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the cookies on the second plate. He continues until all the boxes are empty. Then Serge records the number of cookies on each plate. Again, if the same number appears more than once, it is recorded only once. Prove that Alex's record contains the same number of numbers as Serge's record.
2008 Tournament Of Towns
Solve the system of equations $(n > 2)$



$$\begin{array}{c}\ \sqrt{x_1}+\sqrt{x_2+x_3+\cdots+x_n}=\sqrt{x_2}+\sqrt{x_3+x_4+\cdots+x_n+x_1}=\cdots=\sqrt{x_n}+\sqrt{x_1+x_2+\cdots+x_{n-1}} \end{array}, $$ $$x_1-x_2=1.$$
2008 Tournament Of Towns
A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.
2008 Tournament Of Towns
On the infinite chessboard several rectangular pieces are placed whose sides run along the grid lines. Each two have no squares in common, and each consists of an odd number of squares. Prove that these pieces can be painted in four colours such that two pieces painted in the same colour do not share any boundary points.
2008 Tournament Of Towns
Five distinct positive integers form an arithmetic progression. Can their product be equal to $a^{2008}$ for some positive integer $a$ ?
2008 Tournament Of Towns
A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.
2008 Tournament Of Towns
Space is dissected into congruent cubes. Is it necessarily true that for each cube there exists another cube so that both cubes have a whole face in common?
2008 Tournament Of Towns
There are $N$ piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every $N > 2$ determine which of the players, the first or the second, has a winning strategy.
2008 Tournament Of Towns
Let $ABCD$ be a non-isosceles trapezoid. Define a point $A1$ as intersection of circumcircle of triangle $BCD$ and line $AC$. (Choose $A_1$ distinct from $C$). Points $B_1, C_1, D_1$ are defined in similar way. Prove that $A_1B_1C_1D_1$ is a trapezoid as well.
2008 Tournament Of Towns
Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infinitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.
2008 Tournament Of Towns
A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after



(a)  $30$th attempt?



(b) $25$th attempt?



(Initially, Victor does not know any answer)
2008 Tournament Of Towns
In the convex hexagon $ABCDEF, AB, BC$ and $CD$ are respectively parallel to $DE, EF$ and $FA$. If $AB = DE$, prove that $BC = EF$ and $CD = FA$.
2008 Tournament Of Towns
There are ten congruent segments on a plane. Each point of intersection divides every segment passing through it in the ratio $3:4$. Find the maximum number of points of intersection.
2008 Tournament Of Towns
There are ten cards with the number $a$ on each, ten with $b$ and ten with $c$, where $a, b$ and $c$ are distinct real numbers. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is $0$. Prove that one of $a, b$ and $c$ is $0$.
2008 Tournament Of Towns
Find all positive integers $n$ such that $(n + 1)!$ is divisible by $1! + 2! + ... + n!$.
2008 Tournament Of Towns
Each cell of a $10 \times 10$ board is painted red, blue or white, with exactly twenty of them red. No two adjacent cells are painted in the same colour. A domino consists of two adjacent cells, and it is said to be good if one cell is blue and the other is white.

(a) Prove that it is always possible to cut out $30$ good dominoes from such a board.

(b) Give an example of such a board from which it is possible to cut out $40$ good dominoes.

(c) Give an example of such a board from which it is not possible to cut out more than $30$ good dominoes.
2008 Tournament Of Towns
An integer $N$ is the product of two consecutive integers.

(a) Prove that we can add two digits to the right of this number and obtain a perfect square.

(b) Prove that this can be done in only one way if $N > 12$
2008 Tournament Of Towns
A line parallel to the side $AC$ of triangle $ABC$ cuts the side $AB$ at $K$ and the side $BC$ at $M$. $O$ is the point of intersection of $AM$ and $CK$. If $AK = AO$ and $KM = MC$, prove that $AM = KB$.
2008 Tournament Of Towns
Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?
2008 Tournament Of Towns
Given are finitely many points in the plane, no three on a line. They are painted in four colours, with at least one point of each colour. Prove that there exist three triangles, distinct but not necessarily disjoint, such that the three vertices of each triangle have different colours, and none of them contains a coloured point in its interior.
2008 Tournament Of Towns
Standing in a circle are $99$ girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?
2008 Tournament Of Towns
Do there exist positive integers $a,b,c$ and $d$ such that  $\frac{a}{b} + \frac{c}{d} = 1$ and $\frac{a}{d} + \frac{c}{b} = 2008$ ?
2008 Tournament Of Towns
A convex quadrilateral $ABCD$ has no parallel sides. The angles between the diagonal $AC$ and the four sides are $55^o, 55^o, 19^o$ and $16^o$ in some order. Determine all possible values of the acute angle between $AC$ and $BD$.
2008 Tournament Of Towns
Can it happen that the least common multiple of $1, 2,... , n$ is $2008$ times the least common multiple of $1, 2, ... , m$ for some positive integers $m$ and $n$ ?
2008 Tournament Of Towns
In triangle $ABC, \angle A = 90^o$. $M$ is the midpoint of $BC$ and $H$ is the foot of the altitude from $A$ to $BC$. The line passing through $M$ and perpendicular to $AC$ meets the circumcircle of triangle $AMC$ again at $P$. If $BP$ intersects $AH$ at $K$, prove that $AK = KH$.
2008 Tournament Of Towns
No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.
2008 Tournament Of Towns
We may permute the rows and the columns of the table below.

How may different tables can we generate?



1 2 3 4 5 6 7

7 1 2 3 4 5 6

6 7 1 2 3 4 5

5 6 7 1 2 3 4

4 5 6 7 1 2 3

3 4 5 6 7 1 2

2 3 4 5 6 7 1
2008 Tournament Of Towns
A triangle has an angle of measure $\theta$. It is dissected into several triangles. Is it possible that all angles of the resulting triangles are less than $\theta$, if

(a) $\theta = 70^o$ ?

(b) $\theta = 80^o$ ?
2008 Tournament Of Towns
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?
2008 Tournament Of Towns
A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$.

Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$
2008 Tournament Of Towns
Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are $\alpha, \alpha, \beta$ and $\gamma$  in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also $\alpha, \alpha, \beta$ and $\gamma$   in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.
2008 Tournament Of Towns
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?
2008 Tournament Of Towns
Seated in a circle are $11$ wizards. A different positive integer not exceeding $1000$ is pasted onto the forehead of each. A wizard can see the numbers of the other $10$, but not his own. Simultaneously, each wizard puts up either his left hand or his right hand. Then each declares the number on his forehead at the same time. Is there a strategy on which the wizards can agree beforehand, which allows each of them to make the correct declaration?
2008 Tournament Of Towns
Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.
2009 Tournament Of Towns
Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes?



(3 points)
2009 Tournament Of Towns
There are forty weights: $1, 2, \cdots , 40$ grams. Ten weights with even masses were put on the left pan of a balance. Ten weights with odd masses were put on the right pan of the balance. The left and the right pans are balanced. Prove that one pan contains two weights whose masses differ by exactly $20$ grams.



(4 points)
2009 Tournament Of Towns
A cardboard circular disk of radius $5$ centimeters is placed on the table. While it is possible, Peter puts cardboard squares with side $5$ centimeters outside the disk so that:



(1) one vertex of each square lies on the boundary of the disk;

(2) the squares do not overlap;

(3) each square has a common vertex with the preceding one.



Find how many squares Peter can put on the table, and prove that the first and the last of them must also have a common vertex.



(4 points)
2009 Tournament Of Towns
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?



(5 points for Juniors and 4 points for Seniors)
2009 Tournament Of Towns
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?



(5 points)
2009 Tournament Of Towns
Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves.



(4 points)
2009 Tournament Of Towns
Mike has $1000$ unit cubes. Each has $2$ opposite red faces, $2$ opposite blue faces and $2$ opposite white faces. Mike assembles them into a $10 \times 10 \times 10$ cube. Whenever two unit cubes meet face to face, these two faces have the same colour. Prove that an entire face of the $10 \times 10 \times 10$ cube has the same colour.



(6 points)
2009 Tournament Of Towns
Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$



(6 points)
2009 Tournament Of Towns
Let $ABCD$ be a rhombus. $P$ is a point on side $ BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$



(6 points)
2009 Tournament Of Towns
We have N objects with weights $1, 2,\cdots , N$ grams. We wish to choose two or more of these objects so that the total weight of the chosen objects is equal to average weight of the remaining objects. Prove that



(a) (2 points) if $N + 1$ is a perfect square, then the task is possible;



(b) (6 points) if the task is possible, then $N + 1$ is a perfect square.
2009 Tournament Of Towns
On an infinite chessboard are placed $2009 \  n \times n$ cardboard pieces such that each of them covers exactly $n^2$ cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least $n^2.$



(9 points)
2009 Tournament Of Towns
Anna and Ben decided to visit Archipelago with $2009$ islands. Some pairs of islands are connected by boats which run both ways. Anna and Ben are playing during the trip:



Anna chooses the first island on which they arrive by plane. Then Ben chooses the next island which they could visit. Thereafter, the two take turns choosing an island which they have not yet visited. When they arrive at an island which is connected only to islands they had already visited, whoever's turn to choose next would be the loser. Prove that Anna could always win, regardless of the way Ben played and regardless of the way the islands were connected.



(12 points for Juniors and 10 points for Seniors)
2009 Tournament Of Towns
$A; B; C; D; E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane.



(4 points)
2009 Tournament Of Towns
Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ?



(4 points)
2009 Tournament Of Towns
A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side. Prove that the $2009$-gon with vertices at the images of these reflections also has area $S.$



(4 points)
2009 Tournament Of Towns
A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.



(5 points)
2009 Tournament Of Towns
One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt.



Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate.



Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost.



(4 points)
2009 Tournament Of Towns
A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces.



(6 points)
2009 Tournament Of Towns
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.



(7 points)
2009 Tournament Of Towns
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times  [m]!$



(8 points)
2009 Tournament Of Towns
Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$



(8 points)
2009 Tournament Of Towns
At the entrance to a cave is a rotating round table. On top of the table are $n$ identical barrels, evenly spaced along its circumference. Inside each barrel is a herring either with its head up or its head down. In a move, Ali Baba chooses from $1$ to $n$ of the barrels and turns them upside down. Then the table spins around. When it stops, it is impossible to tell which barrels have been turned over. The cave will open if the heads of the herrings in all $n$ barrels are up or are all down. Determine all values of $n$ for which Ali Baba can open the cave in a finite number of moves.



(11 points)
2009 Tournament Of Towns
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.
2009 Tournament Of Towns
Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?
2009 Tournament Of Towns
Alex is going to make a set of cubical blocks of the same size and to write a digit on each of their faces so that it would be possible to form every $30$-digit integer with these blocks. What is the minimal number of blocks in a set with this property? (The digits $6$ and $9$ do not turn one into another.)
2009 Tournament Of Towns
We increased some positive integer by $10\%$ and obtained a positive integer. Is it possible that in doing so we decreased the sum of digits exactly by $10\%$ ?