Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Tournament of Towns
Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs $+, -, \times$ and a special sign $\pm$. Usage of $\pm$ is equivalent to usage of $+$ and $-$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6\}$, while $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3\}$.

Can Peter construct an expression if the numbers on the blackboard are :

(a) $1, 2, 4$ ? ($2$ points)

(b) any $100$ distinct real numbers ? ($6$ points)
2015 Tournament of Towns
Basil has a melon in a shape of a ball, $20$ in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut, $h$ deep ($h$ is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if

(a) $h = 17$ ? (6 points)

(b) $h = 18$ ? (6 points)
2015 Tournament of Towns
$N$ children no two of the same height stand in a line. The following two-step procedure is applied: first, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of their heights (a group may consist of a single child). Second, the order of children in each group is reversed, so now in each group the children stand in descending order of their heights. Prove that in result of applying this procedure $N - 1$ times the children in the line would stand from the left to the right in descending order of their heights.

(12 points)
2016 Tournament Of Towns
All integers from $1$ to one million are written on a tape in some arbitrary order. Then the tape is cut into pieces containing two consecutive digits each. Prove that these pieces contain all two-digit integers for sure, regardless of the initial order of integers.(4 points)

Alexey Tolpygo
2016 Tournament Of Towns
Do there exist integers $a$ and $b$ such that :

(a) the equation $x^2 + ax + b = 0$ has no real roots, and the equation $\lfloor x^2 \rfloor + ax + b = 0$ has at

least one real root?

(2 points)

(b) the equation $x^2 + 2ax + b$ = 0 has no real roots, and the equation $\lfloor x^2 \rfloor + 2ax + b = 0$ has at

least one real root?

3 points

(By $\lfloor k \rfloor$ we denote the integer part of $k$, that is, the greatest integer not exceeding $k$.)

Alexandr Khrabrov
2016 Tournament Of Towns
Given a square with side $10$. Cut it into $100$ congruent quadrilaterals such that each of them is inscribed into a circle with diameter $\sqrt{3}$. (5 points)



Ilya Bogdanov
2016 Tournament Of Towns
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)

(8 points)



Mikhail Evdokimov
2016 Tournament Of Towns
Let $p$ be a prime integer greater than $10^k$. Pete took some multiple of $p$ and inserted a $k-$digit integer $A$ between two of its neighbouring digits. The resulting integer C was again a multiple of $p$. Pete inserted a $k-$digit integer $B$ between two of neighbouring digits of $C$ belonging to the inserted integer $A$, and the result was again a multiple of $p$. Prove that the integer $B$ can be obtained from the integer $A$ by a permutation of its digits.

(8 points)

Ilya Bogdanov
2016 Tournament Of Towns
Q. An automatic cleaner of the disc shape has passed along a plain floor. For each point of its circular boundary there exists a straight line that has contained this point all the time. Is it necessarily true that the center of the disc stayed on some straight line all the time? ($9$ marks)
2016 Tournament Of Towns
a.) There are $2n+1$ ($n>2$) batteries. We don't know which batteries are good and which are bad but we know that the number of good batteries is greater by $1$ than the number of bad batteries. A lamp uses two batteries, and it works only if both of them are good. What is the least number of attempts sufficient to make the lamp work?



b.) The same problem but the total number of batteries is $2n$ ($n>2$) and the numbers of good and bad batteries are equal.



Proposed by Alexander Shapovalov
2016 Tournament Of Towns
Let $M$ be the midpoint of the base $AC$ of an isosceles $\triangle ABC$. Points $E$ and $F$ on the sides $AB$ and $BC$ respectively are chosen so that $AE \neq CF$ and $\angle FMC = \angle MEF = \alpha$.

Determine $\angle AEM$. (6 points) 



Maxim Prasolov
2016 Tournament Of Towns
There are $64$ towns in a country and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected or not. Our aim is to determine whether it is possible to travel from any town to any other by a sequence of roads. Prove that there is no algorithm which enables us to do so in less than $2016$ questions.



(Proposed by Konstantin Knop)
2016 Tournament Of Towns
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial

on the blackboard has $37$ distinct positive roots. (8 points)



Alexandr Kuznetsov
2016 Tournament Of Towns
Recall that a palindrome is a word which is the same when we read it forward or backward.



(a) We have an infinite number of cards with words $\{abc, bca, cab\}$. A word is made from them in the following way. The initial word is an arbitrary card. At each step we obtain a new word either gluing a card (from the right or from the left) to the existing word or making a cut between any two of its letters and gluing a card between both parts. Is it

possible to obtain a palindrome this way? (4 points)



(b) We have an infinite number of red cards with words $\{abc, bca, cab\}$ and of blue cards with words $\{cba, acb, bac\}$. A palindrome was formed from them in the same way as in part (a). Is it necessarily true that the number of red and blue cards used was equal?  (6 points)



Alexandr Gribalko, Ivan Mitrofanov
2016 Tournament Of Towns
A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for :

(a) $N=3$ (4 points)

(b) $N=4$ (6 points)



Alexandr Berdnikov
2016 Tournament Of Towns
On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$  with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.
2016 Tournament Of Towns
On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point  such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.
2016 Tournament Of Towns
Rectangle $p*q,$ where $p,q$ are relatively coprime  positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right  cuts triangles from some squares.Find sum of perimeters of all such  triangles.
2016 Tournament Of Towns
30 masters and 30 juniors came onto tennis players meeting .Each master played with one master and 15 juniors while each junior played with one junior and 15 masters.Prove that one can find two masters and two juniors such that these masters played with each other ,juniors -with each other ,each of two masters played with at least one of  two  juniors and each of two  juniors played with at least one  of two masters.
2016 Tournament Of Towns
In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values  of 12th edge.
2016 Tournament Of Towns
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find  the smallest possible value of $N $.
2016 Tournament Of Towns
$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies?



(N. Chernyatevya)



(Translated from here.)
2016 Tournament Of Towns
A natural number is written in each cell of an $8 \times 8$ board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than $32$?



(N. Chernyatyev)



(Translated from here.)
2016 Tournament Of Towns
The quadrilateral $ABCD$ is inscribed in circle $\Omega$ with center $O$, not lying on either of the diagonals. Suppose that the circumcircle of triangle $AOC$ passes through the midpoint of the diagonal $BD$. Prove that the circumcircle of triangle $BOD$ passes through the midpoint of diagonal $AC$.



(A. Zaslavsky)



(Translated from here.)
2016 Tournament Of Towns
There are $2016$ red and $2016$ blue cards each having a number written on it. For some $64$ distinct positive real numbers, it is known that the set of numbers on cards of a particular color happens to be the set of their pairwise sums and the other happens to be the set of their pairwise products. Can we necessarily determine which color corresponds to sum and which to product?



(B. Frenkin)



(Translated from here.)
2016 Tournament Of Towns
Is it possible to cut a square of side $1$ into two parts and rearrange them so that one can cover a circle having diameter greater than $1$?



(Note: any circle with diameter greater than $1$ suffices)



(A. Shapovalov)



(Translated from here.)
2016 Tournament Of Towns
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?



(Anant Mudgal)



(Translated from here.)
2016 Tournament Of Towns
Several frogs are sitting on the real line at distinct integer points. In each move, one of them can take a $1$-jump towards the right as long as they are still in on distinct points. We calculate the number of ways they can make $N$ moves in this way for a positive integer $N$. Prove that if the jumps were all towards the left, we will still get the same number of ways.



(F. Petrov)



(Translated from here.)
2017 Tournament Of Towns
A chess tournament had 10 participants. Each round, the participants split into pairs, and

each pair played a game. In total, each participant played with every other participant

exactly once, and in at least half of the games both the players were from the same town.

Prove that during each round there was a game played by two participants from the same

town.



(Boris Frenkin)
2017 Tournament Of Towns
From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all

original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the

cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original

numbers).

a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points)

b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points)



(Alexey Tolpygo)
2017 Tournament Of Towns
All the sides of the convex hexagon $ABCDEF$ are equal. In addition, $AD = BE = CF$.

Prove that a circle can be inscribed into this hexagon.



(Boyan Obukhov)
2017 Tournament Of Towns
There is a set of control weights, each of them weighs a non-integer number of grams. Any

integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control

weights are on one balance pan, and the measured weight on the other pan).What is the

least possible number of the control weights?



(Alexandr Shapovalov)
2017 Tournament Of Towns
A grasshopper can jump along a checkered strip for $8, 9$ or $10$ cells in any direction. A

natural number $n$ is called jumpable if the grasshopper can start from some cell of a strip of

length $n$ and visit every cell exactly once. Find at least one non-jumpable number $n > 50$.



(Egor Bakaev)
2017 Tournament Of Towns
$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially

stick out from the chessboard but the center of each domino must be strictly inside the

chessboard (not on its border). Place on the chessboard in such a way:

a) at least $40$ dominoes, (3 points)

b) at least $41$ dominoes, (3 points)

c) more than $41$ dominoes. (6 points)



(Mikhail Evdokimov)
2018 Tournament Of Towns
Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive?    (4 points)



Boris Frenkin
2018 Tournament Of Towns
Aladdin has several gold ingots, and sometimes he asks the Genie to give him more. The Genie first adds a thousand ingots, but then takes half of the total number. Could it be possible that after asking the Genie for gold ten times, the number of Aladdin’s gold ingots increased, assuming that each time the Genie took half, he took an integer number of ingots? (5 points)



Alexandr Perepechko
2018 Tournament Of Towns
Do there exist 2018 positive irreducible fractions, each with a different denominator, so that the denominator of the difference of any two (after reducing the fraction) is less than the denominator of any of the initial 2018 fractions? (6 points)



Maxim Didin
2018 Tournament Of Towns
Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points)



Egor Bakaev
2018 Tournament Of Towns
There are 100 houses in the street, divided into 50 pairs. In each pair houses are right across the street one from another. On the right side of the street the houses have even numbers, while the houses on the left side have odd numbers. On both sides of the street the numbers increase from the beginning to the end of the street, but are not necessarily consecutive (some numbers may be omitted). For each house on the right side of the street, the difference between its number and the number of the opposite house was computed, and it turned out that all these values were different. Let $n$ be the greatest number of a house on this street. Find the smallest possible value of $n$. (8 points)



Maxim Didin
2018 Tournament Of Towns
In the land of knights (who always tell the truth) and liars (who always lie), 10 people sit at a round table, each at a vertex of an inscribed regular 10-gon, at least one of them is a liar. A traveler can stand at any point outside the table and ask the people: ”What is the distance from me to the nearest liar at the table?” After that each person at the table gives him an answer. What is the minimal number of questions the traveler has to ask to determine which people at the table are liars? (Both the people at the table and the traveler should be considered as points, and everyone can compute the distance between any two points)  (10 points)



Maxim Didin
2018 Tournament Of Towns
You are in a strange land and you don’t know the language. You know that ”!” and ”?” stand for addition and subtraction, but you don’t know which is which. Each of these two symbols can be written between two arguments, but for subtraction you don’t know if the left argument is subtracted from the right or vice versa. So, for instance, a?b could mean any of a − b, b − a, and a + b. You don’t know how to write any numbers, but variables and parenthesis work as usual. Given two arguments a and b, how can you write an expression that equals 20a − 18b? (12 points)



Nikolay Belukhov
2019 Tournament Of Towns
Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers?



(Alexandr Shapovalov)
2019 Tournament Of Towns
Consider 2n+1 coins lying in a circle. At the beginning, all the coins are heads up. Moving clockwise, 2n+1 flips are performed: one coin is flipped, the next coin is skipped, the next coin is flipped, the next two coins are skipped, the next coin is flipped,the next three coins are skipped and so on, until finally 2n coins are skipped and the next coin is flipped.Prove that at the end of this procedure,exactly one coin is heads down.
2019 Tournament Of Towns
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le  n^2$.



(Boris Frenkin)
2019 Tournament Of Towns
Isosceles triangles with a fixed angle $\alpha$ at the vertex opposite to the base are being inscribed into a rectangle $ABCD$ so that this vertex lies on the side $BC$ and the vertices of the base lie on the sides $AB$ and $CD$. Prove that the midpoints of the bases of all such triangles coincide.



(Igor Zhizhilkin)
2019 Tournament Of Towns
A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.
2019 Tournament Of Towns
The King gives the following task to his two wizards. The First Wizard should choose $7$ distinct positive integers with total sum $100$ and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand.



(Mikhail Evdokimov)
2019 Tournament Of Towns
$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position.



(Sergey Dorichenko)
2019 Tournament Of Towns
Two equal non-intersecting wooden disks, one gray and one black, are glued to a plane. A triangle with one gray side and one black side can be moved along the plane so that the disks remain outside the triangle, while the colored sides of the triangle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that the line that contains the bisector of the angle between the gray and black sides always passes through some fixed point of the plane.



(Egor Bakaev, Pavel Kozhevnikov, Vladimir Rastorguev)  (Senior version here)
2019 Tournament Of Towns
Each segment whose endpoints are the vertices of a given regular $100$-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the $100$-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to $1$. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result?



(Ilya Bogdanov)
2019 Tournament Of Towns
One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible?



(Alexandr Gribalko)
2019 Tournament Of Towns
Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?
2019 Tournament Of Towns
There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is $0$?



(Maxim Didin)
2019 Tournament Of Towns
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to $1, 1$ and $2$, respectively. Determine the length of this hexagon's side.



(Mikhail Evdokimov)
2019 Tournament Of Towns
Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$?



(Boris Frenkin)
2019 Tournament Of Towns
Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed.



(Nairi Sedrakyan)
2019 Tournament Of Towns
A magician and his assistant are performing the following trick. There is a row of $13$ empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistant knows which boxes contain coins. The magician returns and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects four boxes, which are then simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always successfully perform the trick.



(Igor Zhizhilkin)



junior version posted here
2019 Tournament Of Towns
Consider a sequence of positive integers with total sum $2019$  such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence?



(Alexandr Shapovalov)