Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Switzerland Team Selection Test
Two circles intersect at points $M$ and $N$. Let $A$ be a point on the first circle, distinct from $M,N$. The lines $AM$ and $AN$ meet the second circle again at $B$ and $C$, respectively. Prove that the tangent to the first circle at $A$ is parallel to $BC$.
1999 Switzerland Team Selection Test
Can the set $\{1,2,...,33\}$ be partitioned into $11$ three-element sets, in each of which one element equals the sum of the other two?
1999 Switzerland Team Selection Test
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.
1999 Switzerland Team Selection Test
Find all real solutions $(x,y,z)$ of the system $\frac{4x^2}{1+4x^2}= y,\frac{4y^2}{1+4y^2}= z, \frac{4z^2}{1+4z^2}= x$.
1999 Switzerland Team Selection Test
In a rectangle $ABCD, M$ and $N$ are the midpoints of $AD$ and $BC$ respectively and $P$ is a point on line $CD$. The line $PM$ meets $AC$ at $Q$. Prove that MN bisects the angle $\angle QNP$.
1999 Switzerland Team Selection Test
Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.
1999 Switzerland Team Selection Test
A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.
1999 Switzerland Team Selection Test
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with

$\sum_{k=1}^{n}a_k = 96, \sum_{k=1}^{n}a_k^2 = 144,\sum_{k=1}^{n}a_k^3 = 216$
1999 Switzerland Team Selection Test
Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients.

Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$
1999 Switzerland Team Selection Test
Prove that the product of five consecutive positive integers cannot be a perfect square.
1998 Switzerland Team Selection Test
A function $f : R -\{0\} \to  R$ has the following properties:

(i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$,

(ii) $f$ takes the value $\frac12$ at least once.

Determine $f(-1)$.

Prove that $f$ is a periodic function
1998 Switzerland Team Selection Test
Find all nonnegative integer solutions $(x,y,z)$ of the equation

$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
1998 Switzerland Team Selection Test
Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.
1998 Switzerland Team Selection Test
Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.
1998 Switzerland Team Selection Test
Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.
1998 Switzerland Team Selection Test
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
1998 Switzerland Team Selection Test
Consider an $n\times n$ matrix whose entry at the intersection of the $i$-th row and the $j-$th column equals $i+ j -1$. What is the largest possible value of the product of $n$ entries of the matrix, no two of which are in the same row or column?
1998 Switzerland Team Selection Test
Let $\vartriangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $X,Y,Z$ respectively. Prove that $XY \cdot YZ\cdot ZX \ge XB\cdot YC\cdot ZA$.
1998 Switzerland Team Selection Test
If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$

.
1998 Switzerland Team Selection Test
5. Let $f : R \to R$ be a function that satisfies for all $x \in R$

(i) $| f(x)| \le 1$, and

(ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$

Prove that $f$ is a periodic function
1997 Switzerland Team Selection Test
1. A finite sequence of integers $a_0,a_1,...,a_n$ is called quadratic if $|a_k -a_{k-1}| = k^2$

for $n\geq k\geq1$.

(a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence

with $a_0 = b$ and $a_n =c$.

(b) Find the smallest natural number $n$ for which there exists a quadratic sequence

with $a_0 = 0$ and $a_n = 1997$
1997 Switzerland Team Selection Test
2. Let ABCD be a convex quadrilateral. Find the necessary and sufficient condition

for the existence of point P inside the quadrilateral such that the triangles

ABP,BCP,CDP,DAP have the same area
1997 Switzerland Team Selection Test
3. A 6×6 square has been tiled by 18 dominoes. Show that there exists a line that

divides the square into two parts, each of which is also tiled by dominoes
1997 Switzerland Team Selection Test
4. Let $v$ and $w$ be two randomly chosen roots of the equation $z^{1997} -1 = 0$ (all roots are equiprobable). Find the probability that $\sqrt{2+\sqrt{3}}\le |u+w|$
TST Round 1
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of

$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$Israel
TST Round 1
Let $n$ and $k$ be positive integers satisfying $k\leq2n^2$. Lee and Sunny play a game with a $2n\times2n$ grid paper. First, Lee writes a non-negative real number no greater than $1$ in each of the cells, so that the sum of all numbers on the paper is $k$. Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most $1$. There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.)

Let $M$ be the number of pieces. Lee wants to maximize $M$, while Sunny wants to minimize $M$. Find the value of $M$ when Lee and Sunny both play optimally.
TST Round 1
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that $$O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.$$(A disk is assumed to contain its boundary.)
TST Round 1
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and $$\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k$$for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
TST Round 1
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
TST Round 1
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and  $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $. Prove that for any $ m \in \{1,2, \dots, 107\} $, the arithmetic mean of the following numbers $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}} $$is greater than or equal to $ \frac{M}{N} $
TST Round 1
Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$Prove that $ A $ is a power of $ 2 $.
TST Round 1
Let $ a,b,c,d $ be four non-negative reals such that $ a+b+c+d = 4 $. Prove that $$ a\sqrt{3a+b+c}+b\sqrt{3b+c+d}+c\sqrt{3c+d+a}+d\sqrt{3d+a+b} \ge 4\sqrt{5} $$
TST Round 1
Given a scalene triangle $ \triangle ABC $. $ B', C' $ are points lie on the rays $ \overrightarrow{AB}, \overrightarrow{AC}  $ such that $ \overline{AB'} = \overline{AC}, \overline{AC'} = \overline{AB} $. Now, for an arbitrary point $ P $ in the plane. Let $ Q $ be the reflection point of $ P $ w.r.t $ \overline{BC} $. The intersections of $ \odot{\left(BB'P\right)} $ and $ \odot{\left(CC'P\right)} $ is $ P' $ and the intersections of $ \odot{\left(BB'Q\right)} $ and $ \odot{\left(CC'Q\right)} $ is $ Q' $. Suppose that $ O, O' $ are circumcenters of $ \triangle{ABC}, \triangle{AB'C'} $ Show that



1. $ O', P', Q' $ are colinear



2. $  \overline{O'P'} \cdot  \overline{O'Q'} = \overline{OA}^{2} $
TST Round 1
For postive integers $k,n$, let

$$f_k(n)=\sum_{m\mid n,m>0}m^k$$Find all pairs of positive integer $(a,b)$ such that $f_a(n)\mid f_b(n)$ for every positive integer $n$.
TST Round 1
Given $a,b,c,d>0$, prove that:

$$\sum_{cyc}\frac{c}{a+2b}+\sum_{cyc}\frac{a+2b}{c}\geq 8(\frac{(a+b+c+d)^2}{ab+ac+ad+bc+bd+cd}-1),$$where $\sum_{cyc}f(a,b,c,d)=f(a,b,c,d)+f(d,a,b,c)+f(c,d,a,b)+f(b,c,d,a)$.
TST Round 1
Suppose function $f:[0,\infty)\to[0,\infty)$ satisfies

(1)$\forall x,y \geq 0,$ we have $f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2})$;

(2)$\forall 0 \leq x \leq 1, f(x) \leq 2016$.

Prove that $f(x)\leq x^2$ for all $x\geq 0$.
TST Round 1
Circles $O_1$ and $O_2$ intersects at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. We join $AB$ to intersect $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume that $H$ is an arbitrary point on the line segment $AF$. We join $HE$ and extend it to intersect circle $O_1$ at point $G$, and join $BG$ and extend it to intersect the extended line of $AC$ at point $D$.

Prove that $\frac{AH}{HF}=\frac{AC}{CD}$.
TST Round 1
Is it possible to divide $\mathbb{N}$ into six disjoint sets $A_1, A_2, A_3, A_4, A_5, A_6$, such that $x,y,z$ are not in the same set if $x+2y=5z$?
TST Round 1
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.
TST Round 1
Prove that for positive reals $a$, $b$, $c$ we have $$ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. $$
TST Round 1
Determine whether there exist ten sets $A_1$, $A_2$, $\dots$, $A_{10}$ such that

(i) each set is of the form $\{a,b,c\}$, where $a \in \{1,2,3\}$, $b \in \{4,5,6\}$, $c \in \{7,8,9\}$,

(ii) no two sets are the same,

(iii) if the ten sets are arranged in a circle $(A_1, A_2, \dots, A_{10})$, then any two adjacent sets have no common element, but any two non-adjacent sets intersect. (Note: $A_{10}$ is adjacent to $A_1$.)
TST Round 1
Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.
TST Round 1
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R} \setminus \{ 0 \}$ such that

$$(f(x))^2f(2y)+(f(y))^2f(2x)=2f(x)f(y)f(x+y)$$

for all $x,y\in\mathbb{Q}$
TST Round 2
Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds

$$\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)$$where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.
TST Round 2
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.



$Proposed \;  by \; Croatia$
TST Round 2
Let $ABCD$ be a convex quadrilateral with pairwise distinct side lengths such that $AC\perp BD$. Let $O_1,O_2$ be the circumcenters of $\Delta ABD, \Delta CBD$, respectively. Show that $AO_2, CO_1$, the Euler line of $\Delta ABC$ and the Euler line of $\Delta ADC$ are concurrent.



(Remark: The Euler line of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)



Proposed by usjl
TST Round 2
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
TST Round 2
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.

Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.

Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.



(Hungary)
TST Round 2
Prove that for any positive reals $ a,b,c,d $ with $ a+b+c+d = 4 $, we have $$ \sum\limits_{cyc}{\frac{3a^3}{a^2+ab+b^2}}+\sum\limits_{cyc}{\frac{2ab}{a+b}} \ge 8 $$
TST Round 2
Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that

$$\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.$$
TST Round 2
Let $ABC$ be a triangle with circumcircle $\Omega$, circumcenter $O$ and orthocenter $H$. Let $S$ lie on $\Omega$ and $P$ lie on $BC$ such that $\angle ASP=90^\circ$, line $SH$ intersects the circumcircle of $\triangle APS$ at $X\neq S$. Suppose $OP$ intersects $CA,AB$ at $Q,R$, respectively, $QY,RZ$ are the altitude of $\triangle AQR$. Prove that $X,Y,Z$ are collinear.



Proposed by Shuang-Yen Lee
TST Round 2
There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
TST Round 2
Find all tuples of positive integers $(a,b,c)$ such that

$$a^b+b^c+c^a=a^c+b^a+c^b$$
TST Round 2
Find all function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that



$f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)$



for all integer $x,y$
TST Round 2
Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero.

Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$.



Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$
TST Round 2
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.
TST Round 2
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1  = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$



Prove that $u_n = v_n.$
TST Round 3
Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has

$$ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. $$
TST Round 3
A city is a point on the plane. Suppose there are $n\geq 2$ cities. Suppose that for each city $X$, there is another city $N(X)$ that is strictly closer to $X$ than all the other cities. The government builds a road connecting each city $X$ and its $N(X)$; no other roads have been built. Suppose we know that, starting from any city, we can reach any other city through a series of road.



We call a city $Y$ suburban if it is $N(X)$ for some city $X$. Show that there are at least $(n-2)/4$ suburban cities.



Proposed by usjl.
TST Round 3
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
TST Round 3
Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th  powers.



Canada
TST Round 3
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.



(Australia)
TST Round 3
For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime.  Find the minimal possible value of the maximum of all numbers.
TST Round 3
Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$.



Show that $FG$ is perpendicular to $IO$.
TST Round 3
In an $n\times{n}$ grid, there are some cats living in each cell (the number of cats in a cell must be a non-negative integer). Every midnight, the manager chooses one cell:

(a) The number of cats living in the chosen cell must be greater than or equal to the number of neighboring cells of the chosen cell.

(b) For every neighboring cell of the chosen cell, the manager moves one cat from the chosen cell to the neighboring cell.

(Two cells are called "neighboring" if they share a common side, e.g. there are only $2$ neighboring cells for a cell in the corner of the grid)

Find the minimum number of cats living in the whole grid, such that the manager is able to do infinitely many times of this process.
TST Round 3
Choose a rational point $P_0(x_p,y_p)$ arbitrary on ellipse $C:x^2+2y^2=2098$. Define $P_1,P_2,\cdots$ recursively by the following rules:



$(1)$ Choose a lattice point $Q_i=(x_i,y_i)\notin C$ such that $|x_i|<50$ and $|y_i|<50$.

$(2)$ Line $P_iQ_i$ intersects $C$ at another point $P_{i+1}$.



Prove that for any point $P_0$ we can choose suitable points $Q_0,Q_1,\cdots$ such that $\exists k\in\mathbb{N}\cup\{0\}$, $\overline{OP_k}^2=2017$.
TST Round 3
Let $n$ be a positive integer. Find the number of odd coefficients of the polynomial $(x^2-x+1)^n$.
TST Round 3
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:

$(i)$ $a_0=a_n=1$;

$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;

$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;

$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and  $a_i$ divides $a_{i-1}+a_{i+1}$.

Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.
TST Round 3
Let $x,y$ be the positive real numbers with $x+y=1$, and $n$ be the positive integer with $n\ge2$. Prove that

$$\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}$$
TST Round 3
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.
TST Round 3
Consider a $6 \times 6$ grid. Define a diagonal to be the six squares whose coordinates $(i,j)$ ($1 \le i,j \le 6)$ satisfy $i-j \equiv k \pmod 6$ for some $k=0,1,\dots,5$. Hence there are six diagonals.



Determine if it is possible to fill it with the numbers $1,2,\dots,36$ (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.
TST Round 3
Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
2018 Tajikistan Team Selection Test
Problem 1. Let ω be the incircle of triangle ABC which is tangent to BC,CA,AB at points D,E,F, respectively. The altitudes of triangle DEF with respect to E,F meet AB,AC at points X,Y, respectively. Prove that the second intersection of the circumcircles of triangles AEX,AFY lies on the circle ω.
2018 Tajikistan Team Selection Test
Problem 2. Prove that for every n≥3, there exists a convex polygon with n sides, such that one can divide it into n-2 triangles that are all similar, but pairwise non-congruent.
2018 Tajikistan Team Selection Test
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression:

(b-a)(b^3+3a^3)
2018 Tajikistan Team Selection Test
Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that:

a/(1-x)+b/(1-y)=1

Prove that:

∛ay+∛bx≤1.
2018 Tajikistan Team Selection Test
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that:

n|a_i b_i-a_j b_j
2018 Tajikistan Team Selection Test
Problem 6. Let H be orthocenter of an acute-angled triangle ABC. Points E,F are on the segments AB,AC respectively, such that BE=BH,CF=CH. The lines EH,FH meet BC in X,Y respectively. Draw the perpendicular HZ from H to EF. Prove that the circumcircle of triangle XYZ is tangent to the circle with diameter BC.
2018 Tajikistan Team Selection Test
Problem 7. On the board, Sabir writes 10 consecutive numbers. For each number, Salim writes the sum of its digits on his paper, and Sabrina writes the number of its divisors on her paper. Is it possible for Sabrina’s 10 numbers to be exactly the same as Salim’s 10 numbers in some order? (the repetitions of the numbers should also be the same)
2018 Tajikistan Team Selection Test
Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows:

K_0=1

K_1 (x_1 )=〖x_1〗^2

K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n )

Prove that:

K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )
2018 Tajikistan Team Selection Test
Problem 9. The numbers 1,2,…,〖97〗^2 are written in the cells of a 97×97 board. In the center of each cell, there is a tower with the height equal to the number of that cell. Is it possible to see the top of any tower from the top of any other tower? (one point A can see the other point B, iff there is no other point on the segment AB).
2014 Tajikistan Team Selection Test
Given the polynomial $p(x) = x^2 + x - 70$, do there exist integers $0<m<n$, so that $p(m)$ is divisible by $n$ and $p(m+1)$ is divisible by $n+1$?



Proposed by Nairy Sedrakyan
2014 Tajikistan Team Selection Test
Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$.



Proposed by Nairy Sedrakyan