Olympiad problems

2016 Turkey EGMO TST, 5

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

2018 JBMO TST-Turkey, 8

Tags: algebra , triangle inequality , positive real numbers , inequalities , Turkey

Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

2014 Turkey Team Selection Test, 3

Tags: geometry , combinatorics , TST , combinatorial geometry , Turkey

At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.

2017 Turkey EGMO TST, 4

Tags: Turkey , EGMO , TST , geometry , contest problem

2021 JBMO TST - Turkey, 4

Tags: inequalities , 2-variable inequality , algebra , Turkey

Let $x,y,z$ be real numbers such that $$\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1$$ Find the maximum value of the expression $$x^3+2y$$

2015 Turkey Team Selection Test, 1

Tags: Turkey , TST , 2015 , number theory

Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.

2021 Turkey Team Selection Test, 2

Tags: Turkey , combinatorics

In a school with some students, for any three student, there exists at least one student who are friends with all these three students.(Friendships are mutual) For any friends $A$ and $B$, any two of their common friends are also friends with each other. It's not possible to partition these students into two groups, such that every student in each group are friends with all the students in the other gruop. Prove that any two people who aren't friends with each other, has the same number of common friends.(Each person is a friend of him/herself.)

2025 Turkey Team Selection Test, 1

Tags: graph theory , combinatorics , edge coloring , Turkey

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

2021 Turkey Team Selection Test, 6

Tags: algebra , Turkey , equation , n-variable equation , construction

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

2019 Turkey EGMO TST, 2

Tags: Turkey , minimum value , inequalities , three variable inequality , algebra

Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$. Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$

2020 Turkey Junior National Olympiad, 4

Tags: combinatorics , combinatorics proposed , Turkey

There are dwarves in a forest and each one of them owns exactly 3 hats which are numbered with numbers $1, 2, \dots 28$. Three hats of a dwarf are numbered with different numbers and there are 3 festivals in this forest in a day. In the first festival, each dwarf wears the hat which has the smallest value, in the second festival, each dwarf wears the hat which has the second smallest value and in the final festival each dwarf wears the hat which has the biggest value. After that, it is realized that there is no dwarf pair such that both of two dwarves wear the same value in at least two festivals. Find the maximum value of number of dwarves.

2025 Turkey Team Selection Test, 9

Tags: number theory , number theory with sequences , modular arithmetic , modular congruences , Turkey

Let $n$ be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these $n$ sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by $n$?

2015 Turkey Team Selection Test, 6

Tags: Turkey , TST , 2015 , number theory

Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.

2022 Dutch BxMO TST, 5

Tags: Turkey , combinatorics

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

2020 Turkey MO (2nd round), 3

Tags: inequalities , inequalities proposed , Turkey

If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

2018 Turkey MO (2nd Round), 3

Tags: Sequences , algebra , number theory , Turkey , Mobius function

A sequence $a_1,a_2,\dots$ satisfy $$ \sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10}, $$ for every $n\in\mathbb{N}$. Let $c$ be a positive integer. Prove that, for every positive integer $n$, $$ \frac{c^{a_n}-c^{a_{n-1}}}{n} $$ is an integer.

2017 Turkey Team Selection Test, 5

Tags: Turkey , TST , algebra , inequalities

For all positive real numbers $a,b,c$ with $a+b+c=3$, show that $$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$

2015 Turkey Team Selection Test, 2

Tags: Turkey , TST , 2015 , combinatorics

There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$?

2020 Turkey MO (2nd round), 1

Tags: combinatorics , Turkey , Divisibility

Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.

2016 EGMO TST Turkey, 5

Tags: combinatorics , Periodic sequence , Turkey , algebra , Sequence

A sequence $a_1, a_2, \ldots $ consisting of $1$'s and $0$'s satisfies for all $k>2016$ that \[ a_k=0 \quad \Longleftrightarrow \quad a_{k-1}+a_{k-2}+\cdots+a_{k-2016}>23. \] Prove that there exist positive integers $N$ and $T$ such that $a_k=a_{k+T}$ for all $k>N$.

2016 Turkey Team Selection Test, 3

Tags: inequalities , Turkey

Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

2021 JBMO TST - Turkey, 2

Tags: number theory solved , Turkey

For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$ is an integer?

2024 Turkey EGMO TST, 1

Tags: geometry , circumcircle , incentre , Turkey , Egmo tst , 2024

Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.

2018 Turkey Junior National Olympiad, 3

Tags: geometry , circumcircle , Turkey , Junior

In an acute $ABC$ triangle which has a circumcircle center called $O$, there is a line that perpendiculars to $AO$ line cuts $[AB]$ and $[AC]$ respectively on $D$ and $E$ points. There is a point called $K$ that is different from $AO$ and $BC$'s junction point on $[BC]$. $AK$ line cuts the circumcircle of $ADE$ on $L$ that is different from $A$. $M$ is the symmetry point of $A$ according to $DE$ line. Prove that $K$,$L$,$M$,$O$ are circular.

2025 Turkey Team Selection Test, 2

Tags: number theory , Turkey , arithmetic sequence

For all positive integers $n$, the function $\gamma: \mathbb{Z}^+ \to \mathbb{Z}_{\geq 0}$ is defined as, $\gamma(1) = 0$ and for all $n > 1$, if the prime factorization of $n$ is $n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k},$ then $\gamma(n) = \alpha_1 + \alpha_2 + \dots + \alpha_k$. We have an arithmetic sequence $X = \{x_i\}_{i=1}^{\infty}$. If for a positive integer $a > 1$, the sequence $\{ \gamma(a^{x_i} -1) \}$ is also an arithmetic sequence, show that the sequence $X$ has to be constant.