This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 114

2022/2023 Tournament of Towns, P2
Tags: Tournament of Towns , number theory , palindromes
Does there exist a natural number that can be represented as the product of two numeric palindromes in more than $100{}$ ways?

2022/2023 Tournament of Towns, P4
Tags: combinatorics , board , Tournament of Towns
Let $n>1$ be an integer. A rook stands in one of the cells of an infinite chessboard that is initially all white. Each move of the rook is exactly $n{}$ cells in a single direction, either vertically or horizontally, and causes the $n{}$ cells passed over by the rook to be painted black. After several such moves, without visiting any cell twice, the rook returns to its starting cell, with the resulting black cells forming a closed path. Prove that the number of white cells inside the black path gives a remainder of $1{}$ when divided by $n{}$.

2022/2023 Tournament of Towns, P2
Tags: geometry , Tournament of Towns
A big circle is inscribed in a rhombus, each of two smaller circles touches two sides of the rhombus and the big circle as shown in the figure on the right. Prove that the four dashed lines spanning the points where the circles touch the rhombus as shown in the figure make up a square.

2019 Tournament Of Towns, 1
Tags: combinatorics , Sum , consecutive , Tournament of Towns , ToT
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)

Kvant 2021, M2674
Tags: combinatorics , Tournament of Towns , Process , Kvant
Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$. [i]Mikhail Lukin[/i]

2020/2021 Tournament of Towns, P4
Tags: geometry , Tournament of Towns
The sides of a triangle are divided by the angle bisectors into two segments each. Is it always possible to form two triangles from the obtained six segments? [i]Lev Emelyanov[/i]

2022/2023 Tournament of Towns, P3
Tags: Tournament of Towns , geometry , pentagon , Angle Chasing
A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

Kvant 2021, M2650
Tags: number theory , Tournament of Towns
For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

2020/2021 Tournament of Towns, P1
Tags: geometry , Tournament of Towns
Is it possible to select 100 points on a circle so that there are exactly 1000 right triangles with the vertices at selected points? [i]Sergey Dvoryaninov[/i]

2022/2023 Tournament of Towns, P4
Tags: geometry , Tournament of Towns
Consider an acute non-isosceles triangle. In a single step it is allowed to cut any one of the available triangles into two triangles along its median. Is it possible that after a finite number of cuttings all triangles will be isosceles? [i]Proposed by E. Bakaev[/i]

2020/2021 Tournament of Towns, P5
Tags: combinatorics , Tournament of Towns
The director of a Zoo has bought eight elephants numbered by $1, 2, \ldots , 8$. He has forgotten their masses but he remembers that each elephant starting with the third one has the mass equal to the sum of the masses of two preceding ones. Suddenly the director hears a rumor that one of the elephants has lost his mass. How can the director perform two weightings on balancing scales without weights to either find this elephant or make sure that this was just a rumor? (It is known that no elephant gained mass and no more than one elephant lost mass.) [i]Alexandr Gribalko[/i]

2020/2021 Tournament of Towns, P5
Tags: number theory , Tournament of Towns
Do there exist 100 positive distinct integers such that a cube of one of them equals the sum of the cubes of all the others? [i]Mikhail Evdokimov[/i]

2022/2023 Tournament of Towns, P5
Tags: combinatorics , game , Tournament of Towns
Alice has 8 coins. She knows for sure only that 7 of these coins are genuine and weigh the same, while the remaining one is counterfeit and is either heavier or lighter than any of the other 7. Bob has a balance scale. The scale shows which plate is heavier but does not show by how much. For each measurement, Alice pays Bob beforehand a fee of one coin. If a genuine coin has been paid, Bob tells Alice the correct weighing outcome, but if a counterfeit coin has been paid, he gives a random answer. Alice wants to identify 5 genuine coins and not to give any of these coins to Bob. Can Alice achieve this result for sure?

2020/2021 Tournament of Towns, P3
Tags: combinatorics , game , Tournament of Towns
There are $n{}$ stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which $n{}$ does the first player have a strategy so that he wins no matter how the other player plays? [i]Fedor Ivlev[/i]

Kvant 2021, M2679
Tags: game , combinatorics , Tournament of Towns , Kvant
The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

2022/2023 Tournament of Towns, P3
Tags: combinatorics , equal segments , Tournament of Towns
There are 2022 marked points on a straight line so that every two adjacent points are the same distance apart. Half of all the points are coloured red and the other half are coloured blue. Can the sum of the lengths of all the segments with a red left endpoint and a blue right endpoint be equal to the sum of the lengths of all the segments with a blue left endpoint and a red right endpoint?

2022/2023 Tournament of Towns, P3
Tags: geometry , Tournament of Towns
Consider two concentric circles $\Omega$ and $\omega$. Chord $AD$ of the circle $\Omega$ is tangent to $\omega$. Inside the minor disk segment $AD$ of $\Omega$, an arbitrary point $P{}$ is selected. The tangent lines drawn from the point $P{}$ to the circle $\omega$ intersect the major arc $AD$ of the circle $\Omega$ at points $B{}$ and $C{}$. The line segments $BD$ and $AC$ intersect at the point $Q{}$. Prove that the line segment $PQ$ passes through the midpoint of line segment $AD$. [i]Note.[/i] A circle together with its interior is called a disk, and a chord $XY$ of the circle divides the disk into disk segments, a minor disk segment $XY$ (the one of smaller area) and a major disk segment $XY$.

2021/2022 Tournament of Towns, P3
Tags: combinatorics , board , Tournament of Towns
In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take? [i]Ivan Yashchenko[/i]

2020/2021 Tournament of Towns, P2
Tags: geometry , algebra , polynomial , Vieta , Tournament of Towns , ToT , Baron Munchausen
Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?

2021/2022 Tournament of Towns, P7
Tags: combinatorics , combinatorial geometry , Tournament of Towns
A checkered square of size $2\times2$ is covered by two triangles. Is it necessarily true that: [list=a] [*]at least one of its four cells is fully covered by one of the triangles; [*]some square of size $1\times1$ can be placed into one of these triangles? [/list] [i]Alexandr Shapovalov[/i]

2020/2021 Tournament of Towns, P4
Tags: algebra , roots , Tournament of Towns
It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation $\lfloor x^2\rfloor+px+q=0$ with $p\neq 0$ to have more than 100 roots? [i]Alexey Tolpygo[/i]

2020/2021 Tournament of Towns, P5
Tags: combinatorics , Tournament of Towns
There are 101 coins in a circle, each weights 10g or 11g. Prove that there exists a coin such that the total weight of the $k{}$ coins to its left is equal to the total weight of the $k{}$ coins to its right where a) $k = 50$ and b) $k = 49$. [i]Alexandr Gribalko[/i]

2020/2021 Tournament of Towns, P4
Tags: geometry , Tournament of Towns
There is an equilateral triangle with side $d{}$ and a point $P{}$ such that the distances from $P{}$ to the vertices of the triangle are positive numbers $a, b, c$. Prove that there exist a point $Q{}$ and an equilateral triangle with side $a{}$, such that the distances from $Q{}$ to the vertices of this triangle are $b, c, d$. [i]Alexandr Evnin[/i]

2020/2021 Tournament of Towns, P5
Tags: combinatorics , combinatorial geometry , Tournament of Towns
Does there exist a rectangle which can be cut into a hundred rectangles such that all of them are similar to the original one but no two are congruent? [i]Mikhail Murashkin[/i]

2020/2021 Tournament of Towns, P3
Tags: geometry , Tournament of Towns
There is an equilateral triangle $ABC$. Let $E, F$ and $K$ be points such that $E{}$ lies on side $AB$, $F{}$ lies on the side $AC$, $K{}$ lies on the extension of side $AB$ and $AE = CF = BK$. Let $P{}$ be the midpoint of the segment $EF$. Prove that the angle $KPC$ is right. [i]Vladimir Rastorguev[/i]