Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Tournament Of Towns
Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane.



(Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version here)

noteThere was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.
2019 Tournament Of Towns
The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides.

a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$?

b) The same question for a square of area $1/2019$.



(Mikhail Evdokimov)
2019 Tournament Of Towns
For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0  \le  x_1 \le ...  \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards?



(Ilya Bogdanov)
2019 Tournament Of Towns
On the grid plane all possible broken lines with the following properties are constructed:

each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding worm, the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$.



(Ilke Chanakchi, Ralf Schiffler)
2019 Tournament Of Towns
The magician puts out hardly a deck of $52$ cards and announces that $51$ of them will be thrown out of the table, and there will remain three of clubs. The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. At what initial positions of the three of clubs can the success of the focus be guaranteed?
2019 Tournament Of Towns
Let $\omega$ be a circle with the center $O$ and $A$ and $C$ be two different points on $\omega$. For any third point $P$ of the circle let $X$ and $Y$ be the midpoints of the segments $AP$ and $CP$. Finally, let $H$ be the orthocenter (the point of intersection of the altitudes) of the triangle $OXY$ . Prove that the position of the point H does not depend on the choice of $P$.



(Artemiy Sokolov)
2019 Tournament Of Towns
There is a row of $100$ cells each containing a token. For $1$ dollar it is allowed to interchange two neighbouring tokens. Also it is allowed to interchange with no charge any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the reverse order?



(Egor Bakaev)
2019 Tournament Of Towns
There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$?



(Boris Frenkin)
2019 Tournament Of Towns
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.



(Mikhail Evdokimov)
2019 Tournament Of Towns
Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, complexity  of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity

a) not greater than the complexity of $n$.

b) less than the complexity of $n$.



(Boris Frenkin)
2019 Tournament Of Towns
Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$



(Nairi Sedrakyan, Ilya Bogdanov)
2019 Tournament Of Towns
There are 100 visually identical coins of three types: golden, silver and copper. There is at least one coin of each type. Each golden coin weighs 3 grams, each silver coins weighs 2 grams and each copper coin weighs 1 gram. How to find the type of each coin performing no more than 101 measurements on a balance scale with no weights.
2019 Tournament Of Towns
Let $OP$ and $OQ$ be the perpendiculars from the circumcenter $O$ of a triangle $ABC$ to the internal and external bisectors of the angle $B$. Prove that the line$ PQ$ divides the segment connecting midpoints of $CB$ and $AB$ into two equal parts.



(Artemiy Sokolov)
2019 Tournament Of Towns
Let us say that the pair $(m, n)$ of two positive different integers m and n is nice  if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice.



(Yury Markelov)
2019 Tournament Of Towns
Peter has several $100$ ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in $1$ ruble coins. Whenever Peter is buying an expensive book for $100$ rubles or higher he uses only $100$ ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than $100$ rubles) he uses his coins if he has enough, otherwise using a $100$ ruble note.

When the $100$ ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended $5000$ rubles or more?



(Tatiana Kazitsina)
2019 Tournament Of Towns
Peter has a wooden square stamp divided into a grid. He coated some $102$ cells of this grid with black ink. After that, he pressed this stamp $100$ times on a list of paper so that each time just those $102$ cells left a black imprint on the paper. Is it possible that after his actions the imprint on the list is a square $101 \times 101$ such that all the cells except one corner cell are black?



(Alexsandr Gribalko)
2019 Tournament Of Towns
Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.
2019 Tournament Of Towns
An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.
2019 Tournament Of Towns
A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.
2019 Tournament Of Towns
In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?
2019 Tournament Of Towns
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?
2019 Tournament Of Towns
Let $ABC$ be an acute triangle. Suppose the points $A',B',C'$ lie on its sides $BC,AC,AB$ respectively and the segments $AA',BB',CC'$ intersect in a common point $P$ inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point $P$ and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that $P$ is the orthocenter of the triangle $ABC$.



(Grigory Galperin)
2019 Tournament Of Towns
Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.
2019 Tournament Of Towns
The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is

a) tangential (circumscribed),

b) cyclic (inscribed).



(Nairi Sedrakyan)
2019 Tournament Of Towns
A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube.

a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions.

b) What is the maximum size of a regular subset that does exist for sure?



(Nikita Gladkov, Alexandr Zimin)
2019 Tournament Of Towns
We color some positive integers $(1,2,...,n)$ with color red, such that any triple of red numbers $(a,b,c)$(not necessarily distincts) if $a(b-c)$ is multiple of $n$ then $b=c$. Prove that the quantity of red numbers is less than or equal to $\varphi(n)$.
2020 Tournament Of Towns
The Quadrumland map is a 6 × 6 square where each square cell is either a kingdom or a disputed territory. There are 27 kingdoms and 9 disputed territories. Each disputed territory is claimed by those and only those kingdoms that are neighbouring with it (adjacent by an edge or a vertex). Is it possible that for each disputed territory the numbers of claims are different?



You can discuss your solutions here
2020 Tournament Of Towns
$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$



I'm posting this problem for people to discuss
2020 Tournament Of Towns
Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$.



Egor Bakaev
2020 Tournament Of Towns
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.



Alexandr Yuran
2020 Tournament Of Towns
On the 8×8 chessboard there are two identical markers in the squares a1 and c3. Alice and Bob in turn make the following moves (the first move is Alice’s): a player picks any marker and moves it horizontally to the right or vertically upwards through any number of squares. The aim of each player is to get tothe square h8. Which player has a winning

strategy no matter what does his opponent? (There may be only one marker on a square,the markers may not go through each other.)



The 8x8 chessboard consists of columns lettered a to h from left to right and rows numbered 1-8 from bottom to top
2020 Tournament Of Towns
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ?



Mikhail Evdokimov
2020 Tournament Of Towns
Three legendary knights are fighting against a multiheaded dragon.

Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads?



Alexey Zaslavsky
2020 Tournament Of Towns
Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where

a) $N = 19$,

b) $N = 20$ ?



Mikhail Malkin
2020 Tournament Of Towns
For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)?



Maxim Didin
2020 Tournament Of Towns
Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle.



Alexandr Yuran
2020 Tournament Of Towns
Alice has a deck of $36$ cards, $4$ suits of $9$ cards each. She picks any $18$ cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bob gains a point. What is the maximum number of points he can guarantee regardless of Alice’s actions?



Mikhail Evdokimov
2020 Tournament Of Towns
Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ?



Ivan Mitrofanov
2020 Tournament Of Towns
Is it possible to fill a $40 \times 41$ table with integers so that each integer equals the number of adjacent (by an edge) cells with the same integer?



Alexandr Gribalko
2020 Tournament Of Towns
Alice asserts that after her recent visit to Addis-Ababa she now has spent the New Year inside every possible hemisphere of Earth except one. What is the minimal number of places where Alice has spent the New Year?



Note: we consider places of spending the New Year to be points on the sphere. A point on the border of a hemisphere does not lie inside the hemisphere.



Ilya Dumansky, Roman Krutovsky
2020 Tournament Of Towns
There are $41$ letters on a circle, each letter is $A$ or $B$. It is allowed to replace $ABA$ by $B$ and conversely, as well as to replace $BAB$ by $A$ and conversely. Is it necessarily true that it is possible to obtain a circle containing a single letter repeating these operations?



Maxim Didin
2020 Tournament Of Towns
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:

a) in exactly one way;

b) in exactly two ways?



Note: two splittings that differ only in the order of summands are considered to be the same.



Sergey Markelov
2020 Tournament Of Towns
Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle.



Alexey Zaslavsky
2020 Tournament Of Towns
Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ?



Alexey Tolpygo
2020 Tournament Of Towns
Alice had picked positive integers $a, b, c$ and then tried to find positive integers $x, y, z$ such that $a = lcm (x, y)$, $b = lcm(x, z)$, $c = lcm(y, z)$. It so happened that such $x, y, z$ existed and were unique. Alice told this fact to Bob and also told him the numbers $a$ and $b$. Prove that Bob can find $c$.  (Note: lcm = least common multiple.)



Boris Frenkin
2020 Tournament Of Towns
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?



Mikhail Evdokimov
2020 Tournament Of Towns
Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways.  (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot  2 \cdot  . . . \cdot  (2N)$.)



Gleb Pogudin
2020 Tournament Of Towns
Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent.



Maxim Didin
2020 Tournament Of Towns
There are $2n$ consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any $2n$ consecutive integers again.



Alexandr Gribalko
2020 Tournament Of Towns
Consider an infinite white plane divided into square cells. For which $k$ it is possible to paint a positive finite number of cells black so that on each horizontal, vertical and diagonal line of cells there is either exactly $k$ black cells or none at all?



A. Dinev, K. Garov, N Belukhov
2020 Tournament Of Towns
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?



A. Gribalko
2020 Tournament Of Towns
At heights $AA_0, BB_0, CC_0$ of an acute-angled non-equilateral triangle $ABC$, points $A_1, B_1, C_1$ were marked, respectively, so that $AA_1 = BB_1 = CC_1 = R$, where $R$ is the radius of the circumscribed circle of triangle $ABC$. Prove that the center of the circumscribed circle of the triangle $A_1B_1C_1$ coincides with the center of the inscribed circle of triangle $ABC$.



E. Bakaev
2020 Tournament Of Towns
$40$ cells were marked on an infinite chessboard. Is it always possible to find a rectangle that contains $20$ marked cells?



M. Evdokimov
2020 Tournament Of Towns
For an infinite sequence $a_1, a_2,. . .$  denote as it's first derivative is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$).  We call a sequence good if it and all its derivatives consist of positive numbers.

Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.



R. Salimov
2020 Tournament Of Towns
A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere.



A. Zaslavsky
2020 Tournament Of Towns
Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors.

We will call the arrangement of the cubes good  if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ successful  if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$.



I. Bogdanov
2020 Tuymaada Olympiad
For each positive integer $m$ let $t_m$ be the smallest positive integer not dividing $m$. Prove that there are infinitely many positive integers which can not be represented in the form $m + t_m$.



(A. Golovanov)
2020 Tuymaada Olympiad
All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$?



(S. Ivanov, K. Kokhas)
2020 Tuymaada Olympiad
Each edge of a complete graph with $101$ vertices is marked with $1$ or  $-1$. It is known that absolute value of the sum of numbers on all the edges is less than $150$. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero.



(Y. Caro, A. Hansberg, J. Lauri, C. Zarb)
2020 Tuymaada Olympiad
Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.
2020 Tuymaada Olympiad
Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?
2020 Tuymaada Olympiad
$AK$ and $BL$ are altitudes of an acute triangle $ABC$. Point $P$ is chosen on the segment $AK$ so that $LK=LP$. The parallel to $BC$ through $P$ meets the parallel to $PL$ through $B$ at point $Q$. Prove that $\angle AQB = \angle ACB$.



(S. Berlov)
2020 Tuymaada Olympiad
How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$?



(F. Bakharev)
2020 Tuymaada Olympiad
In a horizontal strip $1 \times n$ made of $n$ unit squares the vertices of all squares are marked. The strip is partitioned into parts by segments connecting marked points and not lying on the sides of the strip. The segments can not have common inner points; the upper end of each segment must be either above the lower end or further to the right. Prove that the number of all partitions is divisible by $2^n$. (The partition where no segments are drawn, is counted too.)



(E. Robeva, M. Sun)