Olympiad problems

2022/2023 Tournament of Towns, P1

Is it possible to arrange $36$ distinct numbers in the cells of a $6 \times 6$ table, so that in each $1\times 5$ rectangle (both vertical and horizontal) the sum of the numbers equals $2022$ or $2023$?

Kvant 2021, M2675

Tags: board , Chess rook , Tournament of Towns , combinatorics , Kvant

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

2022/2023 Tournament of Towns, P1

Tags: number theory , factorial , Tournament of Towns

Find the maximum integer $m$ such that $m! \cdot 2022!$ is a factorial of an integer.

2022/2023 Tournament of Towns, P4

Tags: board , combinatorics , Tournament of Towns

In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.

Kvant 2020, M2632

Tags: game , combinatorics , Tournament of Towns

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Kvant 2021, M2653

Tags: combinatorics , Combinatorial Number Theory , Tournament of Towns , Kvant

Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. [i]Nikolay Belukhov[/i]

2021/2022 Tournament of Towns, P1

Tags: combinatorics , algebra , Tournament of Towns

Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

2020/2021 Tournament of Towns, P1

Tags: number theory , composite numbers , Tournament of Towns

The number $2021 = 43 \cdot 47$ is composite. Prove that if we insert any number of digits “8” between 20 and 21 then the number remains composite. [i]Mikhail Evdikomov[/i]

Kvant 2019, M2558

Tags: Line , game , game strategy , combinatorics , Tournament of Towns , ToT , Kvant

$2019$ 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)

2020/2021 Tournament of Towns, P5

Tags: geometry , Tournament of Towns

Let $O{}$ be the circumcenter of an acute triangle $ABC$. Let $M{}$ be the midpoint of $AC$. The straight line $BO$ intersects the altitudes $AA_1{}$ and $CC_1{}$ at the points $H_a$ and $H_c$ respectively. The circumcircles of the triangles $BH_aA$ and $BH_cC$ have a second point of intersection $K{}$. Prove that $K{}$ lies on the straight line $BM$. [i]Mikhail Evdokimov[/i]

2020/2021 Tournament of Towns, P2

Tags: geometry , Tournament of Towns

Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles? [i]The Jury[/i]

2020/2021 Tournament of Towns, P1

Tags: number theory , Tournament of Towns

Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

2015 Tournament of Towns, 2

Tags: 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 [b](a) $k=9$?[/b] [b](b) $k=8$?[/b]

2021/2022 Tournament of Towns, P5

Tags: geometry , Tournament of Towns , Kvant

Let $ABCD$ be a parallelogram and let $P{}$ be a point inside it such that $\angle PDA= \angle PBA$. Let $\omega_1$ be the excircle of $PAB$ opposite to the vertex $A{}$. Let $\omega_2$ be the incircle of the triangle $PCD$. Prove that one of the common tangents of $\omega_1$ and $\omega_2$ is parallel to $AD$. [i]Ivan Frolov[/i]

2020/2021 Tournament of Towns, P6

Tags: number theory , Fibonacci , Tournament of Towns

Find at least one real number $A{}$ such that for any positive integer $n{}$ the distance between $\lceil A^n\rceil$ and the nearest square of an integer is equal to two. [i]Dmitry Krekov[/i]

2020/2021 Tournament of Towns, P2

Tags: algebra , Tournament of Towns

Does there exist a positive integer $n{}$ such that for any real $x{}$ and $y{}$ there exist real numbers $a_1, \ldots , a_n$ satisfying \[x=a_1+\cdots+a_n\text{ and }y=\frac{1}{a_1}+\cdots+\frac{1}{a_n}?\] [i]Artemiy Sokolov[/i]

2020/2021 Tournament of Towns, P7

Tags: combinatorics , combinatorial geometry , Tournament of Towns

An integer $n > 2$ is given. Peter wants to draw $n{}$ arcs of length $\alpha{}$ of great circles on a unit sphere so that they do not intersect each other. Prove that [list=a] [*]for all $\alpha<\pi+2\pi/n$ it is possible; [*]for all $\alpha>\pi+2\pi/n$ it is impossible; [/list] [i]Ilya Bogdanov[/i]

2020/2021 Tournament of Towns, P4

Tags: knights and knaves , combinatorics , Tournament of Towns

A traveler arrived to an island where 50 natives lived. All the natives stood in a circle and each announced firstly the age of his left neighbour, then the age of his right neighbour. Each native is either a knight who told both numbers correctly or a knave who increased one of the numbers by 1 and decreased the other by 1 (on his choice). Is it always possible after that to establish which of the natives are knights and which are knaves? [i]Alexandr Gribalko[/i]

2020/2021 Tournament of Towns, P4

Tags: combinatorics , Tournament of Towns

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

2021/2022 Tournament of Towns, P1

Tags: number theory , Tournament of Towns

Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime.

2016 Tournament Of Towns, 3

Tags: rectangle , combinatorics , combinatorial geometry , geometry , 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.

2020/2021 Tournament of Towns, P1

Tags: geometry , Tournament of Towns

[list=a] [*]A convex pentagon is partitioned into three triangles by nonintersecting diagonals. Is it possible for centroids of these triangles to lie on a common straight line? [*]The same question for a non-convex pentagon. [/list] [i]Alexandr Gribalko[/i]

2022/2023 Tournament of Towns, P6

Tags: combinatorics , Tournament of Towns , Kvant

It is known that among several banknotes of pairwise distinct face values (which are positive integers) there are exactly $N{}$ fakes. In a single test, a detector determines the sum of the face values of all real banknotes in an arbitrary set we have selected. Prove that by using the detector $N{}$ times, all fake banknotes can be identified, if a) $N=2$ and b) $N=3$. [i]Proposed by S. Tokarev[/i]

2021/2022 Tournament of Towns, P2

Tags: combinatorics , Tournament of Towns

On a table there are all 8 possible horizontal bars $1\times3$ such that each $1\times1$ square is either white or gray (see the figure). It is allowed to move them in any direction by any (not necessarily integer) distance. We may not rotate them or turn them over. Is it possible to move the bars so that they do not overlap, all the white points form a polygon bounded by a closed non-self-intersecting broken line and the same is true for all the gray points? [i]Mikhail Ilyinsky[/i]

2021/2022 Tournament of Towns, P4

Tags: geometry , 3D geometry , Locus , Tournament of Towns

Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle. [i]Alexandr Matveev[/i]