Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Tournament of Towns
A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one.

Prove that their duty was at least an hour and a half long.
2012 Tournament of Towns
(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines.

(b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.
2012 Tournament of Towns
Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?
2012 Tournament of Towns
Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are

(a) congruent polygons?

(b) regular polygons?
2012 Tournament of Towns
For a class of $20$ students several field trips were arranged. In each trip at least four students participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/17$-th of all field trips.
2012 Tournament of Towns
Let $C(n)$ be the number of prime divisors of a positive integer $n$.

(a) 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 $S$ finite or infinite?

(b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?
2012 Tournament of Towns
Among $239$ coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.
2012 Tournament of Towns
Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?
2012 Tournament of Towns
Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more 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
In a triangle $ABC$ two points, $C_1$ and $A_1$ are marked on the sides $AB$ and $BC$ respectively (the points do not coincide with the vertices). Let $K$ be the midpoint of $A_1C_1$ and $I$ be the incentre of the triangle $ABC$. Given that the quadrilateral $A_1BC_1I$ is cyclic, prove that the angle $AKC$ is obtuse.
2012 Tournament of Towns
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.

(b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its  centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.
2012 Tournament of Towns
There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most $100$ of these numbers are different.
2013 Tournament of Towns
There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?
2013 Tournament of Towns
There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?
2013 Tournament of Towns
Each of $11$ weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than $35$ grams.
2013 Tournament of Towns
Eight rooks are placed on a $8\times 8$ chessboard, so that no two rooks attack one another.

All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.
2013 Tournament of Towns
In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.
2013 Tournament of Towns
Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard?
2013 Tournament of Towns
Twenty children, ten boys and ten girls, are standing in a line. Each boy counted the number of children standing to the right of him. Each girl counted the number of children standing to the left of her. Prove that the sums of numbers counted by the boys and the girls are the same.
2013 Tournament of Towns
There is a $19\times19$ board. Is it possible to mark some $1\times 1$ squares so that each of $10\times 10$ squares contain different number of marked squares?
2013 Tournament of Towns
On a circle, there are $1000$ nonzero real numbers painted black and white in turn. Each black number is equal to the sum of two white numbers adjacent to it, and each white number is equal to the product of two black numbers adjacent to it. What are the possible values of the total sum of $1000$ numbers?
2013 Tournament of Towns
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.
2013 Tournament of Towns
Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.
2013 Tournament of Towns
Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$.

There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line.

Prove that every two players from the opposite teams will eventually play against each other.
2013 Tournament of Towns
Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.
2013 Tournament of Towns
Each of $100$ stones has a sticker showing its true weight. No two stones weight the same. Mischievous Greg wants to rearrange stickers so that the sum of the numbers on the stickers for any group containing from $1$ to $99$ stones is different from the true weight of this group. Is it always possible?
2013 Tournament of Towns
A quadratic trinomial with integer coefficients is called admissible  if its leading coefficient is $1$, its roots are integers and the absolute values of coefficients do not exceed $2013$. Basil has summed up all admissible quadratic trinomials. Prove that the resulting trinomial has no real roots.
2013 Tournament of Towns
A boy and a girl were sitting on a long bench. Then twenty more children one after another came to sit on the bench, each taking a place between already sitting children. Let us call a girl brave if she sat down between two boys, and let us call a boy brave if he sat down between two girls. It happened, that in the end all girls and boys were sitting in the alternating order. Is it possible to uniquely determine the number of brave children?
2013 Tournament of Towns
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.
2013 Tournament of Towns
Integers $1, 2,...,100$ are written on a circle, not necessarily in that order. Can it be that the absolute value of the dierence between any two adjacent integers is at least $30$ and at most $50$?
2013 Tournament of Towns
On an initially colourless plane three points are chosen and marked in red, blue and yellow.

At each step two points marked in different colours are chosen. Then one more point is painted in the third colour so that these three points form a regular triangle with the vertices coloured clockwise in ''red, blue, yellow". A point already marked may be marked again so that it may have several colours. Prove that for any number of moves all the points containing the same colour lie on the same line.
2013 Tournament of Towns
There are five distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal.

(a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers.

(b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).
2013 Tournament of Towns
The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice.

In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it.

(a)  Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council?

(b)  Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?
2013 Tournament of Towns
In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?
2013 Tournament of Towns
Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
2013 Tournament of Towns
Denote by $(a, b)$ the greatest common divisor of $a$ and $b$.

Let $n$ be a positive integer such that $(n, n + 1) < (n, n + 2) <... < (n,n + 35)$. Prove that $(n, n + 35) < (n,n + 36)$.
2013 Tournament of Towns
Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.
2013 Tournament of Towns
Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).
2013 Tournament of Towns
There are $100$ red, $100$ yellow and $100$ green sticks. One can construct a triangle using any three sticks all of different colours (one red, one yellow and one green). Prove that there is a colour such that one can construct a triangle using any three sticks of this colour.
2013 Tournament of Towns
A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.
2013 Tournament of Towns
Assume that $C$ is a right angle of triangle $ABC$ and $N$ is a midpoint of the semicircle, constructed on $CB$ as on diameter externally. Prove that $AN$ divides the bisector of angle $C$ in half.
2013 Tournament of Towns
There is a $8\times 8$ table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?
2013 Tournament of Towns
A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.
2013 Tournament of Towns
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.
2013 Tournament of Towns
On a table, there are $11$ piles of ten stones each. Pete and Basil play the following game. In turns they take $1, 2$ or $3$ stones at a time: Pete takes stones from any single pile while Basil takes stones from different piles but no more than one from each. Pete moves first. The player who cannot move, loses. Which of the players, Pete or Basil, has a winning strategy?
2013 Tournament of Towns
On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram
2013 Tournament of Towns
Denote by $[a, b]$ the least common multiple of $a$ and $b$.

Let $n$ be a positive integer such that $[n, n + 1] > [n, n + 2] >...> [n, n + 35]$. Prove that $[n, n + 35] > [n,n + 36]$.
2013 Tournament of Towns
A spacecraft landed on an asteroid. It is known that the asteroid is either a ball or a cube. The rover started its route at the landing site and finished it at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover transmitted its spatial coordinates to the spacecraft on the landing site so that the trajectory of the rover movement was known. Can it happen that this information is not suffcient to determine whether the asteroid is a ball or a cube?
2013 Tournament of Towns
Find all positive integers $n$ for which the following statement holds:

For any two polynomials $P(x)$ and $Q(x)$ of degree $n$ there exist monomials $ax^k$ and $bx^{ell}, 0 \le k,\ ell \le n$, such that the graphs of $P(x) + ax^k$ and $Q(x) + bx^{ell}$ have no common points.
2013 Tournament of Towns
Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.
2013 Tournament of Towns
Is it true that every integer is a sum of finite number of cubes of distinct integers?
2013 Tournament of Towns
Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have

(a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ?

(b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?
2013 Tournament of Towns
A closed broken self-intersecting line is drawn in the plane. Each of the links of this line is intersected exactly once and no three links intersect at the same point. Further, there are no self-intersections at the vertices and no two links have a common segment. Can it happen that every point of self-intersection divides both links in halves?
2015 Tournament of Towns
(a) The integers $x$, $x^2$ and $x^3$ begin with the same digit. Does it imply that this digit is $1$?

($2$ points) 



(b) The same question for the integers $x, x^2, x^3, \cdots, x^{2015}$

($3$ points)

.
2015 Tournament of Towns
A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$.

($5$ points)
2015 Tournament of Towns
(a)  A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different.

($2$ points)

(b) A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different?

($6$ points)
2015 Tournament of Towns
A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones.

($8$ points)
2015 Tournament of Towns
Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding $2015$ but all coefficients of their product have absolute values not exceeding $1$?

($10$ points)
2015 Tournament of Towns
An Emperor invited $2015$ wizards to a festival. Each of the wizards knows who of them is good and who is evil, however the Emperor doesn’t know this. A good wizard always tells the truth, while an evil wizard can tell the truth or lie at any moment. The Emperor gives each wizard a card with a single question, maybe different for different wizards, and after that listens to the answers of all wizards which are either “yes” or “no”. Having listened to all the answers, the Emperor expels a single wizard through a magic door which shows if this wizard is good or evil. Then the Emperor makes new cards with questions and repeats the procedure with the remaining wizards, and so on. The Emperor may stop at any moment, and after this the Emperor may expel or not expel a wizard. Prove that the Emperor can expel all the evil wizards having expelled at most one good wizard.

($10$ points)
2015 Tournament of Towns
It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to $3$ edges.)

($10$ points)
2015 Tournament of Towns
From a set of integers $\{1,...,100\}$, $k$ integers were deleted. Is it always possible to choose $k$ distinct integers from the remaining set such that their sum is $100$ if



(a) $k=9$?

(b) $k=8$?
2015 Tournament of Towns
Santa Clause had $n$ sorts of candies, $k$ candies of each sort. He distributed them at random between $k$ gift bags, $n$ candies per a bag and gave a bag to everyone of $k$ children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.
2015 Tournament of Towns
A geometrical progression consists of $37$ positive integers. The first and the last terms are relatively prime numbers. Prove that the $19^{th}$ term of the progression is the $18^{th}$ power of some positive integer.

($3$ points)
2015 Tournament of Towns
A $10 \times 10$ square on a grid is split by $80$ unit grid segments into $20$ polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent.

($6$ points)
2015 Tournament of Towns
Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$.

Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$.

($6$ points)
2015 Tournament of Towns
Let $ABCD$ be a cyclic quadrilateral, $K$ and $N$ be the midpoints of the diagonals and $P$ and $Q$ be points of intersection of the extensions of the opposite sides. Prove that $\angle PKQ + \angle PNQ = 180$.

($7$ points)

.