Olympiad problems

2022/2023 Tournament of Towns, P2

Consider two coprime integers $p{}$ and $q{}$ which are greater than $1{}$ and differ from each other by more than $1{}$. Prove that there exists a positive integer $n{}$ such that \[\text{lcm}(p+n, q+n)<\text{lcm}(p,q).\]

2022/2023 Tournament of Towns, P2

Tags: number theory , Tournament of Towns

The numbers $1, 19, 199, 1999,\ldots$ are written on several cards, one card for each number. [list=a] [*]Is it possible to choose at least three cards so that the sum of the numbers on the chosen cards equals a number in which all digits, except for a single digit, are twos? [*]Suppose you have chosen several cards so that the sum of the numbers on the chosen cards equals a number, all of whose digits are twos, except for a single digit. What can this single different digit be? [/list]

2016 Tournament Of Towns, 1

Tags: Tournament of Towns , algebra , logarithm

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.

2021/2022 Tournament of Towns, P5

Tags: geometry , hexagon , Tournament of Towns

A parallelogram $ABCD$ is split by the diagonal $BD$ into two equal triangles. A regular hexagon is inscribed into the triangle $ABD$ so that two of its consecutive sides lie on $AB$ and $AD$ and one of its vertices lies on $BD$. Another regular hexagon is inscribed into the triangle $CBD{}$ so that two of its consecutive vertices lie on $CB$ and $CD$ and one of its sides lies on $BD$. Which of the hexagons is bigger? [i]Konstantin Knop[/i]

Kvant 2022, M2724

Tags: number theory , Tournament of Towns , sum of digits , Kvant

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

2020/2021 Tournament of Towns, P2

Tags: number theory , Tournament of Towns

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]

2022/2023 Tournament of Towns, P5

Tags: number theory , Tournament of Towns , sum of digits , Kvant

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

2021/2022 Tournament of Towns, P1

Tags: number theory , Tournament of Towns

The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event? [i]Alexey Zaslavsky[/i]

2020/2021 Tournament of Towns, P1

Tags: combinatorics , invariant , Tournament of Towns

In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. [i]Maxim Didin[/i]

Kvant 2022, M2722

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, P1

Tags: algebra , polynomial , Tournament of Towns

Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide? [i]Boris Frenkin[/i]

2022/2023 Tournament of Towns, P1

Tags: algebra , Tournament of Towns

What is the largest possible rational root of the equation $ax^2 + bx + c = 0{}$ where $a, b$ and $c{}$ are positive integers that do not exceed $100{}$?

2022/2023 Tournament of Towns, P5

Tags: combinatorics , dominoes , board , Tournament of Towns

A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is: [list=a] [*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves? [/list]

2021/2022 Tournament of Towns, P3

Tags: combinatorics , coins , Tournament of Towns

A pirate has five purses with 30 coins in each. He knows that one purse contains only gold coins, another one contains only silver coins, the third one contains only bronze coins, and the remaining two ones contain 10 gold, 10 silver and 10 bronze coins each. It is allowed to simultaneously take one or several coins out of any purses (only once), and examine them. What is the minimal number of taken coins that is necessary to determine for sure the content of at least one purse? [i]Mikhail Evdokimov[/i]

2019 Tournament Of Towns, 2

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: combinatorics , Process , Tournament of Towns

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

2020/2021 Tournament of Towns, P4

Tags: geometry , Tournament of Towns

[list=a] [*]Is it possible to split a square into 4 isosceles triangles such that no two are congruent? [*]Is it possible to split an equilateral triangle into 4 isosceles triangles such that no two are congruent? [/list] [i]Vladimir Rastorguev[/i]

2020/2021 Tournament of Towns, P3

Tags: game , combinatorics , Tournament of Towns

Alice and Bob are playing the following game. Each turn Alice suggests an integer and Bob writes down either that number or the sum of that number with all previously written numbers. Is it always possible for Alice to ensure that at some moment among the written numbers there are [list=a] [*]at least a hundred copies of number 5? [*]at least a hundred copies of number 10? [/list] [i]Andrey Arzhantsev[/i]

2022/2023 Tournament of Towns, P3

Tags: combinatorics , geometry , Tournament of Towns , Baron Munchausen

Baron Munchausen claims that he has drawn a polygon and chosen a point inside the polygon in such a way that any line passing through the chosen point divides the polygon into three polygons. Could the Baron’s claim be correct?

Kvant 2020, M2633

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]

2020/2021 Tournament of Towns, P2

Tags: combinatorics , Tournament of Towns

A group of 8 players played several tennis tournaments between themselves using the single-elimination system, that is, the players are randomly split into pairs, the winners split into two pairs that play in semifinals, the winners of semifinals play in the final round. It so happened that after several tournaments each player had played with each other exactly once. Prove that [list=a] [*]each player participated in semifinals more than once; [*]each player participated in at least one final. [/list] [i]Boris Frenkin[/i]

2015 Tournament of Towns, 7

Tags: Tournament of Towns

Santa Clause had $n$ sorts of candies, $k$ candies of each sort. He distributed them at random between $k$ gift bags, $n$ candies per a bag and gave a bag to everyone of $k$ children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.

2021/2022 Tournament of Towns, P4

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]

2020/2021 Tournament of Towns, P3

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]

2021/2022 Tournament of Towns, P5

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]