Olympiad problems

2018 Kyiv Mathematical Festival, 5

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

2019 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , inequalities , BPSQ

Let $a,b,c\ge0$ and $a+b+c\ge3.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

2019 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , combinatorics

99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?

2017 Kyiv Mathematical Festival, 1

Tags: permutations , Kyiv mathematical festival , combinatorics

Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have both of their neighbours remained and exactly one third of the dwarves have only one of their neighbours remained, if the number of the dwarves is a) 6; b) 9?

2015 Kyiv Math Festival, P3

Tags: number theory , positive integers , Sum , Kyiv mathematical festival

Is it true that every positive integer greater than $50$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2018 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , combinatorics

There are $n$ ($n \ge 10$) cards with numbers $1, 2, \ldots, n$ lying in a row on a table, face down, so that the numbers on any adjacent cards differ by at least $5.$ Is it always enough to turn at most $n-5$ cards to determine which of the cards has number $n$? (It is not necessary to turn the card with number $n$.)

2019 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , geometry

Let $ABC$ be an isosceles triangle in which $\angle BAC=120^\circ,$ $D$ be the midpoint of $BC,$ $DE$ be the altitude of triangle $ADC,$ and $M$ be the midpoint of $DE.$ Prove that $BM=3AM.$

2017 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , number theory , Diophantine Equations

Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

2021 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , combinatorics

Frodo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Frodo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains odd number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)

2010 Kyiv Mathematical Festival, 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) $S(3)+S(6)+S(9)+\ldots+S(3000).$

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 $ABC$ ($\angle C=90^\circ$), $\omega_1$ 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 $\omega_2$ divides $AC.$

2009 Kyiv Mathematical Festival, 1

Tags: number theory , sum of divisors , Last digit , Kyiv mathematical festival

Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.

2017 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , geometry , number theory , analytic geometry

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

2016 Kyiv Mathematical Festival, P4

Tags: Kyiv mathematical festival , geometry , circles

Let $H$ be the point of intersection of the altitudes $AD$ and $BE$ of acute triangle $ABC.$ The circles with diameters $AE$ and $BD$ touch at point $L$. Prove that $HL$ is the angle bisector of angle $\angle AHB.$

2010 Kyiv Mathematical Festival, 1

Tags: Kyiv mathematical festival

Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure?

2021 Kyiv Mathematical Festival, 1

Tags: Kyiv mathematical festival , number theory

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

2009 Kyiv Mathematical Festival, 3

Tags: geometry , distance , minimum , Kyiv mathematical festival

Points $A_1,A_2,...,A_n$ are selected from the equilateral triangle with a side that is equal to $1$. Denote by $d_k$ the least distance from $A_k$ to all other selected points. Prove that $d_1^2+...+d_n^2 \le 3,5$.

2017 Kyiv Mathematical Festival, 4

Tags: inequalities , Kyiv mathematical festival

Real numbers $x,y$ are such that $x^2\ge y$ and $y^2\ge x.$ Prove that $\frac{x}{y^2+1}+\frac{y}{x^2+1}\le1.$

2016 Kyiv Mathematical Festival, P5

Tags: Kyiv mathematical festival , circles , geometry

Let $AD$ and $BE$ be the altitudes of acute triangle $ABC.$ The circles with diameters $AD$ and $BE$ intersect at points $S$ and $T$. Prove that $\angle ACS=\angle BCT.$

2015 Kyiv Math Festival, P2

Tags: combinatorics , Kyiv mathematical festival , graph theory

In a company of $6$ sousliks each souslik has $4$ friends. Is it always possible to divide this company into two groups of $3$ sousliks such that in both groups all sousliks are friends?

2017 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , combinatorics

Each cell of a $7\times7$ table is painted with one of several colours. It is known that for any two distinct rows the numbers of colours used to paint them are distinct and for any two distinct columns the numbers of colours used to paint them are distinct.What is the maximum possible number of colours in the table?

2009 Kyiv Mathematical Festival, 6

Tags: inequalities , Sets , maximum , Kyiv mathematical festival

Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.

2016 Kyiv Mathematical Festival, P5

Tags: Kyiv mathematical festival , combinatorics , Digits , Divisibility

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

2009 Kyiv Mathematical Festival, 3

Tags: combinatorics , polygon , Kyiv mathematical festival

Let $AB$ be a segment of a plane. Is it possible to paint the plane in $2009$ colors in such a way that both of the following conditions are satisfied? 1) Every two points of the same color can be connected by a polygonal line. 2) For any point $C$ of $AB$, every $n \in N$ and every $k\in \{1,2,3,...,2009\}$ , there exists a point $D$, painted in $k$-th color such that the length of $CD$ is less than $0,0...01$, where all the zeros after the decimal point are exactly $n$.

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 $2019?$