Olympiad problems

2009 Kyiv Mathematical Festival, 5

Tags: number theory , number theory with sequences , divides , Sequence , Kyiv mathematical festival

The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?

2018 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$

2016 Kyiv Mathematical Festival, P5

Tags: Kyiv mathematical festival , combinatorics

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

2019 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , combinatorics , number theory

Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019^4?$

2015 Kyiv Math Festival, P4

Tags: Kyiv mathematical festival , geometry , perimeter , Circumcenter

Let $O$ be the intersection point of altitudes $AD$ and $BE$ of equilateral triangle $ABC.$ Points $K$ and $L$ are chosen inside segments $AO$ and $BO$ respectively such that line $KL$ bisects the perimeter of triangle $ABC.$ Let $F$ be the intersection point of lines $EK$ and $DL.$ Prove that $O$ is the circumcenter of triangle $DEF.$

2010 Kyiv Mathematical Festival, 4

Tags: invariant , combinatorics proposed , combinatorics , Kyiv mathematical festival

1) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy? 2) The numbers $1,2,3,\ldots,2010$ are written on the blackboard. Two players in turn erase some two numbers and replace them with one number. The first player replaces numbers $a$ and $b$ with $ab-a-b+2$ while the second player replaces them with $ab+a+b.$ The game ends when a single number remains on the blackboard. If this number is smaller than $1\cdot2\cdot3\cdot\ldots\cdot2010$ then the first player wins. Otherwise the second player wins. Which of the players has a winning strategy?

2014 Kyiv Mathematical Festival, 2

Tags: inequalities , inequalities proposed , Kyiv mathematical festival

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$

2018 Kyiv Mathematical Festival, 1

Tags: Kyiv mathematical festival , combinatorics

A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?

2017 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , combinatorics , game

Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after both players made five moves, they exchange hats.The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

2009 Kyiv Mathematical Festival, 2

Tags: algebra , three variable inequality , inequalities , positive , Kyiv mathematical festival

Let $x,y,z$ be positive numebrs such that $x+y+z\le x^3+y^3+z^3$. Is it true that a) $x^2+y^2+z^2 \le x^3+y^3+z^3$ ? b) $x+y+z\le x^2+y^2+z^2$ ?

2014 Kyiv Mathematical Festival, 5

Tags: geometry , geometry proposed , Kyiv mathematical festival

Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.

2015 Kyiv Math Festival, P2

Tags: Kyiv mathematical festival , graph theory , combinatorics

In a company of 7 sousliks each souslik has 4 friends. Is it always possible to find in this company two non-intersecting groups of 3 sousliks each such that in both groups all sousliks are friends?

2021 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , geometry

Let $\omega$ be the circumcircle of a triangle $ABC$ ($AB>AC$), $E$ be the midpoint of the arc $AC$ which does not contain point $B,$ аnd $F$ the midpoint of the arc $AB$ which does not contain point $C.$ Lines $AF$ and $BE$ meet at point $P,$ line $CF$ and $AE$ meet at point $R,$ and the tangent to $\omega$ at point $A$ meets line $BC$ at point $Q.$ Prove that points $P,Q,R$ are collinear. (M. Kurskiy)

2016 Kyiv Mathematical Festival, P3

Tags: Kyiv mathematical festival , combinatorics , games

Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What maximal number of the cells can remain unpainted when the game stops?

2021 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , combinatorics

In several cells of a square grid there live hedgehogs. Every hedgehog multiplies the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers?

2018 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , number theory , Divisibility

Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?

2015 Kyiv Math Festival, P1

Tags: Kyiv mathematical festival , Solve equation , algebra

Solve equation $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

2010 Contests, 2

Tags: Kyiv mathematical festival

Denote by $S(n)$ the sum of digits of integer $n.$ Find 1) $S(3)+S(6)+S(9)+\ldots+S(300);$ 2) $S(3)+S(6)+S(9)+\ldots+S(3000).$

2021 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , algebra , number theory

Is it true that for every $n\ge 2021$ there exist $n$ integer numbers such that the square of each number is equal to the sum of all other numbers, and not all the numbers are equal? (O. Rudenko)

2018 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , number theory

Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$

2010 Kyiv Mathematical Festival, 5

Tags: floor function , number theory , relatively prime , combinatorics proposed , combinatorics , Kyiv mathematical festival

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

2018 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , geometry , tangent circles

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$),\linebreak $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_1$ divides $AB.$

2015 Kyiv Math Festival, P5

Tags: Kyiv mathematical festival , combinatorics

Tom painted round fence which consists of $2n \ge6$ sections in such way that every section is painted in one of four colours. Then he repeats the following while it is possible: he chooses three neighbouring sections of distinct colours and repaints them into the fourth colour. For which $n$ Tom can repaint the fence in such way infinitely many times?

2015 Kyiv Math Festival, P1

Tags: Kyiv mathematical festival , algebra

Prove that there exist infinitely many pairs of real numbers $(x,y)$ such that $\sqrt{1+2x-xy}+\sqrt{1+2y-xy}=2.$

2016 Kyiv Mathematical Festival, P1

Tags: equation , Kyiv mathematical festival , algebra

Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$