Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Tournament of Towns
Baron Munchausen has a set of $50$ coins. The mass of each is a distinct positive integer not

exceeding $100$, and the total mass is even. The Baron claims that it is not possible to divide

the coins into two piles with equal total mass. Can the Baron be right?
2011 Tournament of Towns
Given an integer $n > 1$, prove that there exist distinct positive integers $a, b, c$ and $d$ such

that $a + b = c + d$ and $\frac{a}{b}=\frac{nc}{d}$.
2011 Tournament of Towns
$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped

to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$.

Prove that $FG$ is parallel to $HK$ and $FK = GH$.
2011 Tournament of Towns
Two ants crawl along the sides of the $49$ squares of a $7 * 7$ board. Each ant passes through

all $64$ vertices exactly once and returns to its starting point. What is the smallest possible

number of sides covered by both ants?
2011 Tournament of Towns
In every cell of a square table is a number. The sum of the largest two numbers in each row

is $a$ and the sum of the largest two numbers in each column is b. Prove that $a = b$.
2011 Tournament of Towns
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.
2011 Tournament of Towns
Along a circle are $100$ white points. An integer $k$ is given, where $2 \le k \le 50$. In each move, we choose a block of $k$ adjacent points such that the first and the last are white, and we paint both of them black. For which values of $k$ is it possible for us to paint all $100$ points black after $50$ moves?
2011 Tournament of Towns
Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.
2011 Tournament of Towns
In a country, there are $100$ towns. Some pairs of towns are joined by roads. The roads do not intersect one another except meeting at towns. It is possible to go from any town to any other town by road. Prove that it is possible to pave some of the roads so that the number of paved roads at each town is odd.
2011 Tournament of Towns
In the coordinate space, each of the eight vertices of a rectangular box has integer coordinates. If the volume of the solid is $2011$, prove that the sides of the rectangular box are parallel to the coordinate axes.
2011 Tournament of Towns
(a) Does there exist an innite triangular beam such that two of its cross-sections are similar but not congruent triangles?

(b) Does there exist an innite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?
2011 Tournament of Towns
There are $n$ red sticks and $n$ blue sticks. The sticks of each colour have the same total length, and can be used to construct an $n$-gon. We wish to repaint one stick of each colour in the other colour so that the sticks of each colour can still be used to construct an $n$-gon. Is this always possible if

(a) $n = 3$,

(b) $n > 3$ ?
2011 Tournament of Towns
In the convex quadrilateral $ABCD, BC$ is parallel to $AD$. Two circular arcs $\omega_1$ and $\omega_3$ pass through $A$ and $B$ and are on the same side of $AB$. Two circular arcs $\omega_2$ and $\omega_4$ pass through $C$ and $D$ and are on the same side of $CD$. The measures of $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are $\alpha, \beta,\beta$  and $\alpha$ respectively. If $\omega_1$ and $\omega_2$ are tangent to each other externally, prove that so are $\omega_3$ and $\omega_4$.
2011 Tournament of Towns
Among a group of programmers, every two either know each other or do not know each other. Eleven of them are geniuses. Two companies hire them one at a time, alternately, and may not hire someone already hired by the other company. There are no conditions on which programmer a company may hire in the first round. Thereafter, a company may only hire a programmer who knows another programmer already hired by that company. Is it possible for the company which hires second to hire ten of the geniuses, no matter what the hiring strategy of the other company may be?
2011 Tournament of Towns
$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.
2011 Tournament of Towns
Several guests at a round table are eating from a basket containing $2011$ berries. Going in clockwise direction, each guest has eaten either twice as many berries as or six fewer berries than the next guest. Prove that not all the berries have been eaten.
2011 Tournament of Towns
From the $9 \times 9$ chessboard, all $16$ unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.
2011 Tournament of Towns
The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?
2011 Tournament of Towns
On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at $10$ o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at $11$ o'clock, when did he meet the cart?
2011 Tournament of Towns
An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?
2011 Tournament of Towns
On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.
2011 Tournament of Towns
A balance and a set of pairwise different weights are given. It is known that for any pair of weights from this set put on the left pan of the balance, one can counterbalance them by one or several of the remaining weights put on the right pan. Find the least possible number of weights in the set.
2011 Tournament of Towns
A checkered table consists of $2012$ rows and $k > 2$ columns. A marker is placed in a cell of the left-most column. Two players move the marker in turns. During each move, the player moves the marker by $1$ cell to the right, up or down to a cell that had never been occupied by the marker before. The game is over when any of the players moves the marker to the right-most column. However, whether this player is to win or to lose, the players are advised only when the marker reaches the second column from the right. Can any player secure his win?
2011 Tournament of Towns
Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$
2011 Tournament of Towns
A car goes along a straight highway at the speed of $60$ km per hour. A $100$ meter long fence is standing parallel to the highway. Every second, the passenger of the car measures the angle of vision of the fence. Prove that the sum of all angles measured by him is less than $1100$ degrees.
2011 Tournament of Towns
The vertices of a regular $45$-gon are painted into three colors so that the number of vertices of each color is the same. Prove that three vertices of each color can be selected so that three triangles formed by the chosen vertices of the same color are all equal.
2011 Tournament of Towns
Peter buys a lottery ticket on which he enters an $n$-digit number, none of the digits being $0$. On the draw date, the lottery administrators will reveal an $n\times n$ table, each cell containing one of the digits from $1$ to $9$. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?
2011 Tournament of Towns
In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.
2011 Tournament of Towns
Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?
2011 Tournament of Towns
In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
2011 Tournament of Towns
Pete has marked several (three or more) points in the plane such that all distances between them are different. A pair of marked points $A,B$ will be called unusual if $A$ is the furthest marked point from $B$, and $B$ is the nearest marked point to $A$ (apart from $A$ itself). What is the largest possible number of unusual pairs that Pete can obtain?
2011 Tournament of Towns
In triangle $ABC$, points $A_1,B_1,C_1$ are bases of altitudes from vertices $A,B,C$, and points $C_A,C_B$ are the projections of $C_1$ to $AC$ and $BC$ respectively. Prove that line $C_AC_B$ bisects the segments $C_1A_1$ and $C_1B_1$.
2011 Tournament of Towns
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for

a) $N = 2011$

b) $N = 2012$ ?
2011 Tournament of Towns
We will call a positive integer good  if all its digits are nonzero. A good integer will be called special  if it has at least $k$ digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest $k$ such that any good integer can be turned into any other good integer by such moves?
2011 Tournament of Towns
Prove that the integer $1^1 + 3^3 + 5^5 + .. + (2^n - 1)^{2^n-1}$ is a multiple of $2^n$ but not a multiple of $2^{n+1}$.
2011 Tournament of Towns
$100$ red points divide a blue circle into $100$ arcs such that their lengths are all positive integers from $1$ to $100$ in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.
2011 Tournament of Towns
There are $n$ coins in a row. Two players take turns picking a coin and flipping it. The location of the heads and tails should not repeat. Loses the one who can not make a move. Which of player can always win, no matter how his opponent plays?
2011 Tournament of Towns
$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$.



original wording in RussianНа доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600
2011 Tournament of Towns
Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?
2011 Tournament of Towns
A subset of a student group is called an ideal company if

1) in this subset, all girls are liked by all young men,

2) no one can be added to this subset without violating condition $1$.

In a certain group, $9$ female students and $15$ students study. Warden of the group made a list of all kinds of ideal companies in this group. What is the largest number of companies on this list?
2011 Tournament of Towns
Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.
2011 Tournament of Towns
On the plane there are centrally symmetric convex polygon with area 1 and two his copies (each obtained from a polygon by some parallel transfer). It is known that no point of the plane is not covered by the three polygons at once. Prove that the total area covered by polygons, at least 2.
2012 Tournament of Towns
A treasure is buried under a square of an $8\times 8$ board. Under each other square is a message which indicates the minimum number of steps needed to reach the square with the treasure. Each step takes one from a square to another square sharing a common side. What is the minmum number of squares we must dig up in order to bring up the treasure for sure?
2012 Tournament of Towns
The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?
2012 Tournament of Towns
In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.
2012 Tournament of Towns
Brackets are to be inserted into the expression $10 \div  9 \div  8 \div  7 \div  6 \div  5 \div  4 \div  3 \div  2$ so that the resulting number is an integer.

(a) Determine the maximum value of this integer.

(b) Determine the minimum value of this integer.
2012 Tournament of Towns
RyNo, a little rhinoceros, has $17$ scratch marks on its body. Some are horizontal and the rest are vertical. Some are on the left side and the rest are on the right side. If RyNo rubs one side of its body against a tree, two scratch marks, either both horizontal or both vertical, will disappear from that side. However, at the same time, two new scratch marks, one horizontal and one vertical, will appear on the other side. If there are less than two horizontal and less than two vertical scratch marks on the side being rubbed, then nothing happens. If RyNo continues to rub its body against trees, is it possible that at some point in time, the numbers of horizontal and vertical scratch marks have interchanged on each side of its body?
2012 Tournament of Towns
It is possible to place an even number of pears in a row such that the masses of any two neighbouring pears differ by at most $1$ gram. Prove that it is then possible to put the pears two in a bag and place the bags in a row such that the masses of any two neighbouring bags differ by at most $1$ gram.
2012 Tournament of Towns
One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fifty segments intersect one another?
2012 Tournament of Towns
In a team of guards, each is assigned a different positive integer. For any two guards, the ratio of the two numbers assigned to them is at least $3:1$. A guard assigned the number $n$ is on duty for $n$ days in a row, off duty for $n$ days in a row, back on duty for $n$ days in a row, and so on. The guards need not start their duties on the same day. Is it possible that on any day, at least one in such a team of guards is on duty?
2012 Tournament of Towns
Each entry in an $n\times n$ table is either $+$ or $-$. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are $+$. Prove that this can be accomplished in at most $n$ steps.
2012 Tournament of Towns
Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called interesting  if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.
2012 Tournament of Towns
A bank has one million clients, one of whom is Inspector Gadget. Each client has a unique PIN number consisting of six digits. Dr. Claw has a list of all the clients. He is able to break into the account of any client, choose any $n$ digits of the PIN number and copy them. The n digits he copies from different clients need not be in the same $n$ positions. He can break into the account of each client, but only once. What is the smallest value of $n$ which allows Dr.Claw to determine the complete PIN number of Inspector Gadget?
2012 Tournament of Towns
Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.
2012 Tournament of Towns
Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.
2012 Tournament of Towns
The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.
2012 Tournament of Towns
Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.
2012 Tournament of Towns
A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.
2012 Tournament of Towns
In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the first player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?
2012 Tournament of Towns
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
2012 Tournament of Towns
Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.
2012 Tournament of Towns
Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.
2012 Tournament of Towns
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle  B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.
2012 Tournament of Towns
We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed.

(a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$?

(b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?
2012 Tournament of Towns
Konstantin has a pile of $100$ pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets $100 $ piles each with a single pebble.

(a) Prove that at some point, there are $30$ piles containing a total of exactly $60$ pebbles.

(b) Prove that at some point, there are $20$ piles containing a total of exactly $60$ pebbles.

(c) Prove that Konstantin may proceed in such a way that at no point, there are $19$ piles containing a total of exactly $60$ pebbles.
2012 Tournament of Towns
Five students have the first names Clark, Donald, Jack, Robin and Steve, and have the last names (in a different order) Clarkson, Donaldson, Jackson, Robinson and Stevenson. It is known that Clark is $1$ year older than Clarkson, Donald is $2$ years older than Donaldson, Jack is $3$ years older than Jackson, Robin is $4$ years older than Robinson. Who is older, Steve or Stevenson and what is the difference in their ages?
2012 Tournament of Towns
Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$).

Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$.

Is set $S$ finite or infinite?
2012 Tournament of Towns
A table $10 \times  10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex.
2012 Tournament of Towns
A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.
2012 Tournament of Towns
For a class of $20$ students several field trips were arranged. In each trip at least one student participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/20$-th of all field trips.
2012 Tournament of Towns
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?
2012 Tournament of Towns
Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
2012 Tournament of Towns
Some cells of a $11 \times  11$ table are filled with pluses. It is known that the total number of pluses in the given table and in any of its $2 \times 2$ sub-tables is even. Prove that the total number of pluses on the main diagonal of the given table is also even.

($2 \times 2$ sub-table consists of four adjacent cells, four cells around a common vertex).
2012 Tournament of Towns
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?