Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Tajikistan Team Selection Test
Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality

\begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}
2014 Tajikistan Team Selection Test
In a convex hexagon $ABCDEF$ the diagonals $AD,BE,CF$ intersect at a point $M$. It is known that the triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are acute. It is also known that the quadrilaterals $ABDE,BCEF,CDFA$ have the same area. Prove that the circumcenters of triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are concyclic.



Proposed by Nairy Sedrakyan
2014 Tajikistan Team Selection Test
There are $12$ delegates in a mathematical conference. It is known that every two delegates share a common friend. Prove that there is a delegate who has at least five friends in that conference.



Proposed by Nairy Sedrakyan
2014 Turkey EGMO TST
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $AD$ intersect the circumcircle of $ABC$ for the second time at $E$. Let $P$ be the point symmetric to the point $E$ with respect to the point $D$ and $Q$ be the point of intersection of the lines $CP$ and $AB$. Prove that if $A,C,D,Q$ are concyclic, then the lines $BP$ and $AC$ are perpendicular.
2014 Turkey EGMO TST
$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.
2014 Turkey EGMO TST
Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy;

$$P(n+d(n))=n+d(P(n)) $$for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.
2014 Turkey EGMO TST
Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that;

$$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$
2014 Turkey EGMO TST
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\omega_A$ be a circle drawn outside $ABC$ and tangent to side $BC$ at $A_1$ and tangent to $\omega$ at $A_2$. Let the circles $\omega_B$ and $\omega_C$ and the points $B_1, B_2, C_1, C_2$ are defined similarly. Prove that if the lines $AA_1, BB_1, CC_1$ are concurrent, then the lines $AA_2, BB_2, CC_2$ are also concurrent.
2014 Turkey EGMO TST
For a given integer $n\ge3$, let $S_1, S_2,\ldots,S_m$ be distinct three-element subsets of the set $\{1,2,\ldots,n\}$ such that for each $1\le i,j\le m; i\neq j$ the sets $S_i\cap S_j$ contain exactly one element. Determine the maximal possible value of $m$ for each $n$.
2015 Turkey EGMO TST
$a$ is a real number. Find the all $(x,y)$ real number pairs satisfy;$$y^2=x^3+(a-1)x^2+a^2x$$$$x^2=y^3+(a-1)y^2+a^2y$$
2015 Turkey EGMO TST
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\angle PAD=90^\circ - \angle PBD=\angle CAD$. Prove that $\angle PQB=\angle BAC$, where $Q$ is the intersection point of the lines $PC$ and $AD$.
2015 Turkey EGMO TST
Given a $2015$-tuple $(a_1,a_2,\ldots,a_{2015})$ in each step we choose two indices $1\le k,l\le 2015$ with $a_k$ even and transform the $2015$-tuple into $(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})$. Prove that starting from $(1,2,\ldots,2015)$ in finite number of steps one can reach any permutation of $(1,2,\ldots,2015)$.
2015 Turkey EGMO TST
Find the all $(m,n)$ integer pairs satisfying $m^4+2n^3+1=mn^3+n$.
2015 Turkey EGMO TST
Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as;



$K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$



in the cordinate plane.
2015 Turkey EGMO TST
In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.
2016 Turkey EGMO TST
Prove that

$$ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) $$for all positive real numbers $x, y, z$.
2016 Turkey EGMO TST
In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.
2016 Turkey EGMO TST
Let $X$ be a variable point on the side $BC$ of a triangle $ABC$. Let $B'$ and $C'$ be points on the rays $[XB$ and $[XC$, respectively, satisfying $B'X=BC=C'X$. The line passing through $X$ and parallel to $AB'$ cuts the line $AC$ at $Y$ and the line passing through $X$ and parallel to $AC'$ cuts the line $AB$ at $Z$. Prove that all lines $YZ$ pass through a fixed point as $X$ varies on the line segment $BC$.
2016 Turkey EGMO TST
In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.
2016 Turkey EGMO TST
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 EGMO TST
Prove that for every square-free integer $n>1$, there exists a prime number $p$ and an integer $m$ satisfying

$$ p \mid n \quad \text{and} \quad n \mid p^2+p\cdot m^p. $$
2017 Turkey EGMO TST
Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation:

$3^m5^k=n^3+125$
2017 Turkey EGMO TST
At the beginning there are $2017$ marbles in each of $1000$ boxes. On each move Aybike chooses a box, grabs some of the marbles from that box and delivers them one for each to the boxes she wishes. At least how many moves does Aybike have to make to have different number of marbles in each box?
2017 Turkey EGMO TST
For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$prove that

$$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$
2017 Turkey EGMO TST
On the inside of the triangle $ABC$ a point $P$ is chosen with $\angle BAP = \angle CAP$. If $\left | AB \right |\cdot \left | CP \right |= \left | AC \right |\cdot \left | BP \right |= \left | BC \right |\cdot \left | AP \right |$ , find all possible values of the angle $\angle ABP$.
2017 Turkey EGMO TST
In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?



Note. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.
2017 Turkey EGMO TST
Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.
2018 Turkey EGMO TST
Let $ABCD$ be a cyclic quadrilateral and $w$ be its circumcircle.  For a given point $E$ inside $w$, $DE$ intersects $AB$ at $F$ inside $w$. Let $l$ be a line passes through $E$ and tangent to circle $AEF$. Let $G$ be any point on $l$ and inside the quadrilateral $ABCD$. Show that if $\angle GAD =\angle BAE$ and $\angle GCB + \angle GAB = \angle EAD + \angle AGD +  \angle ABE$ then $BC$, $AD$ and $EG$ are concurrent.
2018 Turkey EGMO TST
Determine  all pairs $(m,n)$ of positive integers such that $m^2+n^2=2018(m-n)$
2018 Turkey EGMO TST
In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
2018 Turkey EGMO TST
There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of $n$.
2018 Turkey EGMO TST
Prove that

$\dfrac {x^2+1}{(x+y)^2+4 (z+1)}+\dfrac {y^2+1}{(y+z)^2+4 (x+1)}+\dfrac {z^2+1}{(z+x)^2+4 (y+1)} \ge \dfrac{1}{2} $

for all positive reals $x,y,z$
2018 Turkey EGMO TST
Let $f:\mathbb{Z}_{+}\rightarrow\mathbb{Z}_{+}$ is one to one and bijective function. Prove that $f(mn)=f (m)f (n)$ if and only if  $lcm (f (m),f (n))=f(lcm(m,n)) $
2019 Turkey EGMO TST
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$.
2019 Turkey EGMO TST
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 $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$
2019 Turkey EGMO TST
Let $\omega$ be the circumcircle of $\Delta ABC$, where $|AB|=|AC|$. Let $D$ be any point on the minor arc $AC$. Let $E$ be the reflection of point $B$ in line $AD$. Let $F$ be the intersection of $\omega$ and line $BE$ and Let $K$ be the intersection of line $AC$ and the tangent at $F$. If line $AB$ intersects line $FD$ at $L$, Show that $K,L,E$ are collinear points
2019 Turkey EGMO TST
Let $\sigma (n)$ shows the number of positive divisors of $n$. Let $s(n)$ be the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n)- \sigma (n) $.
2019 Turkey EGMO TST
Let $D$ be the midpoint of $\overline{BC}$ in $\Delta ABC$. Let $P$ be any point on $\overline{AD}$. If the internal angle bisector of $\angle ABP$ and $\angle ACP$ intersect at $Q$. Prove that, if $BQ \perp QC$, then $Q$ lies on $AD$
2019 Turkey EGMO TST
There are $k$ piles and there are $2019$ stones totally. In every move we split a pile into two or remove one pile. Using finite moves we can reach conclusion that there are $k$ piles left and all of them contain different number of stonws. Find the maximum of $k$.
2020 Turkey EGMO TST
$H$ is the orthocenter of a non-isosceles acute triangle $\triangle ABC$. $M$ is the midpoint of $BC$ and $BB_1, CC_1$ are two altitudes of $\triangle ABC$. $N$ is the midpoint of $B_1C_1$. Prove that $AH$ is tangent to the circumcircle of $\triangle MNH$.
2020 Turkey EGMO TST
$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$
2020 Turkey EGMO TST
There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.
2020 Turkey EGMO TST
Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.
2020 Turkey EGMO TST
$A, B, C, D, E$ points are on $\Gamma$ cycle clockwise. $[AE \cap [CD = \{M\}$ and $[AB \cap [DC = \{N\}$. The line parallels to $EC$ and passes through $M$ intersects with the line parallels to $BC$ and passes through $N$ on $K$. Similarly, the line parallels to $ED$ and passes through $M$ intersects with the line parallels to $BD$ and passes through $N$ on $L$. Show that the lines $LD$ and $KC$ intersect on $\Gamma$.
2020 Turkey EGMO TST
$x,y,z$ are positive real numbers such that:

$$xyz+x+y+z=6$$$$xyz+2xy+yz+zx+z=10$$Find the maximum value of:

$$(xy+1)(yz+1)(zx+1)$$
2021 Turkey Team Selection Test
Let \(n\) be a positive integer. Prove that $$\frac{20 \cdot 5^n-2}{3^n+47}$$is not an integer.
2021 Turkey Team Selection Test
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.)
2021 Turkey Team Selection Test
A point $D$ is taken on the arc $BC$ of the circumcircle of triangle $ABC$ which does not contain $A$. A point $E$ is taken at the intersection of the interior region of the triangles $ABC$ and $ADC$ such that $m(\widehat{ABE})=m(\widehat{BCE})$. Let the circumcircle of the triangle $ADE$ meets the line $AB$ for the second time at $K$. Let $L$ be the intersection of the lines $EK$ and $BC$, $M$ be the intersection of the lines $EC$ and $AD$, $N$ be the intersection of the lines $BM$ and $DL$. Prove that $$m(\widehat{NEL})=m(\widehat{NDE})$$
2021 Turkey Team Selection Test
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 $k$?
2021 Turkey Team Selection Test
In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively.

Prove that the lines $DF$, $EG$ and $MN$ are concurrent.
2021 Turkey Team Selection Test
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 $i=1,2,\cdots ,n$ ?
2021 Turkey Team Selection Test
Given a triangle $ABC$ with the circumcircle $\omega$ and incenter $I$. Let the line pass through the point $I$ and the intersection of exterior angle bisector of $A$ and $\omega$ meets the circumcircle of $IBC$ at $T_A$ for the second time. Define $T_B$ and $T_C$ similarly. Prove that the radius of the circumcircle of the triangle $T_AT_BT_C$ is twice the radius of $\omega$.
2021 Turkey Team Selection Test
Let \(c\) be a real number. For all \(x\) and \(y\) real numbers we have,

$$f(x-f(y))=f(x-y)+c(f(x)-f(y))$$and \(f(x)\) is not constant.

\(a)\) Find all possible values of \(c\).

\(b)\) Can \(f\) be periodic?
2021 Turkey Team Selection Test
For which positive integer couples $(k,n)$, the equality



$\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$



holds?
2020 Turkey Team Selection Test
Find all pairs of $(a,b)$ positive integers satisfying the equation:

$$\frac {a^3+b^3}{ab+4}=2020$$
2020 Turkey Team Selection Test
$A_1A_2A_3A_4$ is a tangential quadrilateral with perimeter $p_1$ and sum of the diagonals $k_1$ .$B_1B_2B_3B_4$ is a tangential quadrilateral with perimeter $p_2$ and sum of the diagonals $k_2$ .Prove that  $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are congruent squares if $$ p_1^2+p_2^2=(k_1+k_2)^2 $$
2020 Turkey Team Selection Test
66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.
2020 Turkey Team Selection Test
Let $Z^+$ be positive integers set. $f:\mathbb{Z^+}\to\mathbb{Z^+}$ is a function and

we show $ f \circ f \circ ...\circ f $ with $f_l$ for all $l\in \mathbb{Z^+}$ where $f$ is repeated $l$ times. Find all $f:\mathbb{Z^+}\to\mathbb{Z^+}$ functions such that $$ (n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019}  $$for all $n\in \mathbb{Z^+}$
2020 Turkey Team Selection Test
There is at least one friend pair in a class of students with different names. Students in an ordered list of some of the students write the names of all their friends who are not currently written on the blackboard, in order. If each student on the list wrote at least one name on the board and the name of each student with at least one friend on the blackboard at the end of the process, call this list a $golden$ $ list$. Prove that there exists a $golden$ $ list$ such that number of students in this list is even.
2020 Turkey Team Selection Test
In a triangle $\triangle ABC$, $D$ and $E$ are respectively on $AB$ and $AC$ such that $DE\parallel BC$. $P$ is the intersection of $BE$ and $CD$. $M$ is the second intersection of $(APD)$ and $(BCD)$ ,  $N$ is the second intersection of $(APE)$ and $(BCE)$. $w$ is the circle passing through $M$ and $N$ and tangent to $BC$. Prove that the lines tangent to $w$ at $M$ and $N$ intersect on $AP$.
2020 Turkey Team Selection Test
$A_1,A_2,B_1,B_2,C_1,C_2$ are points on a circle such that $A_1A_2 \parallel B_1B_2 \parallel C_1C_2 $ . $M$ is a point on same circle $MA_1$ and $B_2C_2$ intersect at $X$  , $MB_1$ and $A_2C_2$ intersect at $Y$, $MC_1$ and $A_2B_2$ intersect at $Z$ .Prove that $X , Y ,Z$ are collinear.
2020 Turkey Team Selection Test
Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$
2020 Turkey Team Selection Test
For $a,n$ positive integers we show number of different integer 10-tuples $ (x_1,x_2,...,x_{10})$ on $ (mod n)$  satistfying $x_1x_2...x_{10}=a (mod n)$ with $f(a,n)$. Let $a,b$ given positive integers ,

a) Prove that there exist a positive integer $c$ such that for all $n\in \mathbb{Z^+}$ $$\frac {f(a,cn)}{f(b,cn)}$$is constant

b) Find all $(a,b)$ pairs such that minumum possible value of $c$ is 27 where $c$ satisfying condition in $(a)$
2019 Turkey Team SeIection Test
In each one of the given $2019$ boxes, there are $2019$ stones numbered as $1,2,...,2019$ with total mass of $1$ kilogram. In all situations satisfying these conditions, if one can pick stones from different boxes with different numbers, with total mass of at least 1 kilogram, in $k$ different ways, what is the maximal of $k$?
2019 Turkey Team SeIection Test
$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$.

$a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite.

$b)$ Find 3 different prime numbers that do not divide any terms of this sequence.
2019 Turkey Team SeIection Test
In a triangle $ABC$, $AB>AC$. The foot of the altitude from $A$ to $BC$ is $D$, the intersection of bisector of $B$ and $AD$ is $K$, the foot of the altitude from $B$ to $CK$ is $M$ and let $BM$ and $AK$ intersect at point $N$. The line through $N$ parallel to $DM$ intersects $AC$ at $T$. Prove that $BM$ is the bisector of angle $\widehat{TBC}$.
2019 Turkey Team SeIection Test
For an integer $n$ with $b$ digits, let a subdivisor of $n$ be a positive number which divides a number obtained by removing the $r$ leftmost digits and the $l$ rightmost digits of $n$ for nonnegative integers $r,l$ with $r+l<b$  (For example, the subdivisors of $143$ are $1$, $2$, $3$, $4$, $7$, $11$, $13$, $14$, $43$, and $143$). For an integer $d$, let $A_d$ be the set of numbers that don't have $d$ as a subdivisor. Find all $d$, such that $A_d$ is finite.
2019 Turkey Team SeIection Test
$P(x)$ is a nonconstant polynomial with real coefficients and its all roots are real numbers. If there exist a $Q(x)$ polynomial with real coefficients that holds the equality for all $x$ real numbers

$(P(x))^{2}=P(Q(x))$,

then prove that all the roots of $P(x)$ are same.
2019 Turkey Team SeIection Test
$k$ is a positive integer,

$R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$

$R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ for $n=2k+1$.

A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from $R_{n}$ with the property that any two connected marbles have different numbers on them, we call it nice labeling.  If each one of the marbles are written on some numbers from $R_{n}$ with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to $0$, we call it precise labeling.

$n\geq{3}$, if every mechanism that is labeled nicely with $R_{n}$, could be labeled precisely with $R_{m}$, what is the minimal value of $m$?
2019 Turkey Team SeIection Test
In a triangle $ABC$ with $\angle ACB = 90^{\circ}$ $D$ is the foot of the altitude of $C$. Let $E$ and $F$ be the reflections of $D$ with respect to $AC$ and $BC$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle {ECB}$ and $\triangle {FCA}$. Show that: $$2O_1O_2=AB$$
2019 Turkey Team SeIection Test
Let $p>2$ be a prime number, $m>1$ and $n$ be positive integers such that $\frac {m^{pn}-1}{m^n-1}$ is a prime number. Show that: $$pn\mid (p-1)^n+1$$
2019 Turkey Team SeIection Test
Let $x, y, z$ be real numbers such that $y\geq 2z \geq 4x$ and $$ 2(x^3+y^3+z^3)+15(xy^2+yz^2+zx^2)\geq 16(x^2y+y^2z+z^2x)+2xyz.$$Prove that: $4x+y\geq 4z$
2018 Turkey Team Selection Test
Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.
2018 Turkey Team Selection Test
Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that

$$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$for all real numbers $x,y$.
2018 Turkey Team Selection Test
A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.
2018 Turkey Team Selection Test
In a non-isosceles acute triangle $ABC$, $D$ is the midpoint of the edge $[BC]$. The points $E$ and $F$ lie on $[AC]$ and $[AB]$, respectively, and the circumcircles of $CDE$ and $AEF$ intersect in $P$ on $[AD]$. The angle bisector from $P$ in triangle $EFP$ intersects $EF$ in $Q$. Prove that the tangent line to the circumcirle of $AQP$ at $A$ is perpendicular to $BC$.
2018 Turkey Team Selection Test
We say that a group of $25$ students is a team if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is special if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.
2018 Turkey Team Selection Test
$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either

$$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$or

$$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$Given $a_{100}\leq a_0$, what is the maximal value of $b_1+b_2+\cdots+b_{100}$?