Found problems: 114
2020/2021 Tournament of Towns, P7
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]
Kvant 2022, M2723
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]
2020/2021 Tournament of Towns, P5
There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells?
[i]Nikolay Chernyatiev[/i]
Kvant 2020, M2631
There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that:
[list=a]
[*]one of its angles is not greater than $60^\circ{}$;
[*]one of its angles is at least $120^\circ$.
[/list]
[i]Maxim Didin[/i]
2015 Tournament of Towns, 6
Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression:
$a)$ If the numbers on the blackboard are $1, 2, 4$;
$b)$ For any collection of $100$ distinct real numbers on a blackboard?
2022/2023 Tournament of Towns, P4
Is it possible to colour all integers greater than $1{}$ in three colours (each integer in one colour, all three colours must be used) so that the colour of the product of any two differently coloured numbers is different from the colour of each of the factors?
2021/2022 Tournament of Towns, P4
A convex $n{}$-gon with $n > 4$ is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the $n{}$-gon.
[i]Maxim Didin[/i]
2021/2022 Tournament of Towns, P6
Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true:
\[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\]
[i]Maxim Didin[/i]
2021/2022 Tournament of Towns, P3
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment?
[i]Maxim Volchkevich[/i]
2021/2022 Tournament of Towns, P2
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]
2020/2021 Tournament of Towns, P3
A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have?
[i]Sergey Tokarev[/i]
2020/2021 Tournament of Towns, P3
Let $M{}$ be the midpoint of the side $BC$ of the triangle $ABC$. The circle $\omega$ passes through $A{}$, touches the line $BC$ at $M{}$, intersects the side $AB$ at the point $D{}$ and the side $AC$ at the point $E{}$. Let $X{}$ and $Y{}$ be the midpoints of $BE$ and $CD$ respectively. Prove that the circumcircle of the triangle $MXY$ touches $\omega$.
[i]Alexey Doledenok[/i]
2021/2022 Tournament of Towns, P6
There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob?
[i]Alexandr Gribalko[/i]
2021/2022 Tournament of Towns, P2
A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds.
[i]Oleg Smirnov[/i]
2022/2023 Tournament of Towns, P6
Peter added a positive integer $M{}$ to a positive integer $N{}$ and noticed that the sum of the digits of the resulting integer is the same as the sum of the digits of $N{}$. Then he added $M{}$ to the result again, and so on. Will Peter eventually get a number with the same digit sum as the number $N{}$ again?
Kvant 2020, M2630
Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.)
[i]Boris Frenkin[/i]
2020/2021 Tournament of Towns, P6
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]
2020/2021 Tournament of Towns, P1
There were $n{}$ positive integers. For each pair of those integers Boris wrote their arithmetic mean onto a blackboard and their geometric mean onto a whiteboard. It so happened that for each pair at least one of those means was integer. Prove that on at least one of the boards all the numbers are integer.
[i]Boris Frenkin[/i]
2020/2021 Tournament of Towns, P7
There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that:
[list=a]
[*]one of its angles is not greater than $60^\circ{}$;
[*]one of its angles is at least $120^\circ$.
[/list]
[i]Maxim Didin[/i]
2021/2022 Tournament of Towns, P5
There were 20 participants in a chess tournament. Each of them played with each other twice: once as white and once as black. Let us say that participant $X{}$ is no weaker than participant $Y{}$ if $X{}$ has won at least the same number of games playing white as $Y{}$ and also has won at least the same number of games playing black as $Y{}$ . Do there exist for sure two participants $A{}$ and $B{}$ such that $A{}$ is not weaker than $B{}$?
[i]Boris Frenkin[/i]
2022/2023 Tournament of Towns, P1
One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).
Kvant 2021, M2676
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, P1
Let us say that a circle intersects a quadrilateral [i]properly[/i] if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly?
[i]Alexandr Perepechko[/i]
2001 Tournament Of Towns, 5
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
Kvant 2021, M2677
There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob?
[i]Alexandr Gribalko[/i]