Olympiad problems

2021 Kyiv Mathematical Festival, 5

Bilbo 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 Bilbo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains even 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 Contests, 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?

2017 Kyiv Mathematical Festival, 5

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 the second player duplicated the move of the first player, they exchange hats. The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

2021 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , geometry

Let $\omega$ be the circumcircle of a triangle $ABC$ (${AB\ne AC}$), $I$ be the incenter, $P$ be the point on $\omega$ for which $\angle API=90^\circ,$ $S$ be the intersection point of lines $AP$ and $BC,$ $W$ be the intersection point of line $AI$ and $\omega.$ Line which passes through point $W$ orthogonally to $AW$ meets $AP$ and $BC$ at points $D$ and $E$ respectively. Prove that $SD=IE.$ (Ye. Azarov)

2019 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , inequalities , BPSQ

Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

2014 Kyiv Mathematical Festival, 4a

Tags: number theory , least common multiple , modular arithmetic , number theory proposed , Kyiv mathematical festival

a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$ b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$

2018 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , combinatorics , geometry , game strategy

A circle is divided by $2018$ 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?

2017 Kyiv Mathematical Festival, 1

Tags: Kyiv mathematical festival , permutations , 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 two new neighbours and exactly one third of the dwarves have only one new neighbour, if the number of the dwarves is a) 9; b) 12?

2019 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , geometry

Let $D$ be the midpoint of the base $BC$ of an isosceles triangle $ABC,$ $E$ be the point at the side $AC$ such that $\angle CDE=60^\circ,$ and $M$ be the midpoint of $DE.$ Prove that $\angle AME=\angle BMD.$

2015 Kyiv Math Festival, P3

Tags: Kyiv mathematical festival , number theory , divisor

Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?

2019 Kyiv Mathematical Festival, 1

Tags: Kyiv mathematical festival , number theory

A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?

2010 Kyiv Mathematical Festival, 3

Tags: geometry , circumcircle , incenter , geometry proposed , Kyiv mathematical festival

Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.

2014 Kyiv Mathematical Festival, 3a

Tags: combinatorics proposed , combinatorics , Kyiv mathematical festival

a) There are 8 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D$ such that pairs of teams $A,B$ and $C,D$ won the same number of games in total. b) There are 25 teams in a Quidditch tournament. Each team plays every other team once without draws. Prove that there exist teams $A,B,C,D,E,F$ such that pairs of teams $A,B,$ $~$ $C,D$ and $E,F$ won the same number of games in total.

2016 Kyiv Mathematical Festival, P2

Tags: Kyiv mathematical festival , combinatorial geometry , combinatorics

1) Is it possible to place five circles on the plane in such way that each circle has exactly 5 common points with other circles? 2) Is it possible to place five circles on the plane in such way that each circle has exactly 6 common points with other circles? 3) Is it possible to place five circles on the plane in such way that each circle has exactly 7 common points with other circles?

2019 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , combinatorics

There were $2n,$ $n\ge2,$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same non-zero number of points?

2021 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , algebra , number theory

Find all collections of $63$ 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)

2009 Kyiv Mathematical Festival, 5

Tags: algebra , Sequence , recurrence relation , recursion , inequalities , Kyiv mathematical festival

a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be? b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

2021 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , geometry

Let $AD$ be the altitude, $AE$ be the median, and $O$ be the circumcenter of a triangle $ABC.$ Points $X$ and $Y$ are selected inside the triangle such that $\angle BAX=\angle CAY,$ $OX\perp AX,$ and $OY\perp AY.$ Prove that points $D,E,X,Y$ are concyclic. (M. Kurskiy)

2017 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , geometry , Triangle

A triangle $ABC$ is given. Let $D$ be a point on the extension of the segment $AB$ beyond $A$ such that $AD=BC,$ and $E$ be a point on the extension of the segment $BC$ beyond $B$ such that $BE=AC.$ Prove that the circumcircle of the triangle $DEB$ passes through the incenter of the triangle $ABC.$

2018 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

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 $ABC$ ($\angle C=90^\circ$). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$

2015 Kyiv Math Festival, P3

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

Is it true that every positive integer greater than $100$ is a sum of $4$ positive integers such that each two of them have a common divisor greater than $1$?

2017 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , geometry , circles

A point $C$ is marked on a chord $AB$ of a circle $\omega.$ Let $D$ be the midpoint of $AC,$ and $O$ be the center of the circle $\omega.$ The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F.$ Prove that the circumcircle of the triangle $CEF$ touches $AB.$

2019 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , combinatorics

There were $n\ge2$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same number of points?

2021 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , inequalities , algebra

Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)