Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 Moldova Team Selection Test
The circles $W_1, W_2, W_3$ in the plane are pairwise externally tangent to each other. Let $P_1$ be the point of tangency between circles $W_1$ and $W_3$, and let $P_2$ be the point of tangency between circles $W_2$ and $W_3$. $A$ and $B$, both different from $P_1$ and $P_2$, are points on $W_3$ such that $AB$ is a diameter of $W_3$. Line $AP_1$ intersects $W_1$ again at $X$, line $BP_2$ intersects $W_2$ again at $Y$, and lines $AP_2$ and $BP_1$ intersect at $Z$. Prove that $X, Y$, and $Z$ are collinear.
2002 Moldova Team Selection Test
The sequence Pn (x), n ∈ N of polynomials is defined as follows:

P0 (x) = x, P1 (x) = 4x³ + 3x

Pn+1 (x) = (4x² + 2)Pn (x) − Pn−1 (x), for all n ≥ 1

For every positive integer m, we consider the set A(m) = { Pn (m) | n ∈ N }. Show that the sets A(m) and A(m+4) have no common elements.
2002 Moldova Team Selection Test
Positive numbers $\alpha ,\beta , x_1, x_2,\ldots, x_n$ ($n \geq 1$) satisfy $x_1+x_2+\cdots+x_n = 1$. Prove that

$$\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}.$$

Note. $x_{n+1}=x_1$.
2002 Moldova Team Selection Test
Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an

integer. Find the smallest $k$ for which the set A can be partitioned into two subsets

having the same number of elements, the same sum of elements, the same sum

of the squares of elements, and the same sum of the cubes of elements.
2002 Moldova Team Selection Test
A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$,  respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.
2002 Moldova Team Selection Test
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.
2002 Moldova Team Selection Test
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.
2002 Moldova Team Selection Test
Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.
2002 Moldova Team Selection Test
A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.
2002 Moldova Team Selection Test
Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).
2011 Mongolia Team Selection Test
Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \equiv a (\mod m)$.



I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?
2011 Mongolia Team Selection Test
Mongolia TST 2011 Test 1 #2



Let $p$ be a prime number. Prove that:



$\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$



(proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)



Note: I believe they meant to say $p>2$ as well.
2011 Mongolia Team Selection Test
We are given an acute triangle $ABC$. Let $(w,I)$ be the inscribed circle of $ABC$, $(\Omega,O)$ be the circumscribed circle of $ABC$, and $A_0$ be the midpoint of altitude $AH$. $w$ touches $BC$ at point $D$. $A_0 D$ and $w$ intersect at point $P$, and the perpendicular from $I$ to $A_0 D$ intersects $BC$ at the point $M$. $MR$ and $MS$ lines touch $\Omega$ at $R$ and $S$ respectively [note: I am not entirely sure of what is meant by this, but I am pretty sure it means draw the tangents to $\Omega$ from $M$]. Prove that the points $R,P,D,S$ are concyclic.

(proposed by E. Enkzaya, inspired by Vietnamese olympiad problem)
2011 Mongolia Team Selection Test
A group of the pupils in a class are called dominant if any other pupil from the class has a friend in the group. If it is known that there exist at least 100 dominant groups, prove that there exists at least one more dominant group.



(proposed by B. Batbayasgalan, inspired by Komal problem)
2011 Mongolia Team Selection Test
Let $ABC$ be a scalene triangle. The inscribed circle of $ABC$ touches the sides $BC$, $CA$, and $AB$ at the points $A_1$, $B_1$, $C_1$ respectively. Let $I$ be the incenter, $O$ be the circumcenter, and lines $OI$ and $BC$ meet at point $D$. The perpendicular line from $A_1$ to $B_1 C_1$ intersects $AD$ at point $E$. Prove that $B_1 C_1$ passes through the midpoint of $EA_1$.
2011 Mongolia Team Selection Test
Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?
2011 Mongolia Team Selection Test
Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there

a) exist

b) exist infinitely many

$x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$.



(proposed by B. Bayarjargal)
2011 Mongolia Team Selection Test
Given a triangle $ABC$, the internal and external bisectors of angle $A$ intersect $BC$ at points $D$ and $E$ respectively. Let $F$ be the point (different from $A$) where line $AC$ intersects the circle $w$ with diameter $DE$. Finally, draw the tangent at $A$ to the circumcircle of triangle $ABF$, and let it hit $w$ at $A$ and $G$. Prove that $AF=AG$.
2011 Mongolia Team Selection Test
Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa)



(proposed by B. Batbaysgalan, folklore).
2011 Mongolia Team Selection Test
Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that

$\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$



(proposed by B. Amarsanaa, folklore)
2011 Mongolia Team Selection Test
Let $r$ be a given positive integer. Is is true that for every $r$-colouring of the natural numbers there exists a monochromatic solution of the equation $x+y=3z$?



(proposed by B. Batbaysgalan, folklore)
2011 Mongolia Team Selection Test
Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square.



(proposed by G. Batzaya, folklore)
2008 Mongolia Team Selection Test
Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE = \angle ABC$.
2008 Mongolia Team Selection Test
Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a + b - c + d) | (ac + bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m + c^m d^n$ is composite number
2008 Mongolia Team Selection Test
Given a circumscribed trapezium $ ABCD$ with circumcircle $ \omega$ and 2 parallel sides $ AD,BC$ ($ BC<AD$). Tangent line of circle $ \omega$ at the point $ C$ meets with the line $ AD$ at point $ P$. $ PE$ is another tangent line of circle $ \omega$ and $ E\in\omega$. The line $ BP$ meets circle $ \omega$ at point $ K$. The line passing through the point $ C$ paralel to $ AB$ intersects with $ AE$ and $ AK$ at points $ N$ and $ M$ respectively. Prove that $ M$ is midpoint of segment $ CN$.
2008 Mongolia Team Selection Test
Find all function $ f: R^+ \rightarrow R^+$ such that for any $ x,y,z \in R^+$ such that $ x+y \ge z$ , $ f(x+y-z) +f(2\sqrt{xz})+f(2\sqrt{yz}) = f(x+y+z)$
2008 Mongolia Team Selection Test
Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.
2008 Mongolia Team Selection Test
Find the maximum number $ C$ such that for any nonnegative $ x,y,z$ the inequality

$ x^3 + y^3 + z^3 + C(xy^2 + yz^2 + zx^2) \ge (C + 1)(x^2 y + y^2 z + z^2 x)$ holds.
2008 Mongolia Team Selection Test
Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n = 0}^\infty$ such that $ a_n = 2^n + a$. Prove that the sequence $ \{a_n\}_{n = 0}^\infty$ has finitely number of square of integer.
2008 Mongolia Team Selection Test
The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD = P$ and $ AC\cap B'D = Q$ then prove that $ PQ \perp AC$
2008 Mongolia Team Selection Test
Given positive integers $ m,n > 1$. Prove that the equation

$ (x + 1)^n + (x + 2)^n + ... + (x + m)^n = (y + 1)^{2n} + (y + 2)^{2n} + ... + (y + m)^{2n}$ has finitely number of solutions $ x,y \in N$
2008 Mongolia Team Selection Test
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?
2008 Mongolia Team Selection Test
Let $ a_1,a_2,...,a_n$ is permutaion of $ 1,2,...,n$. For this permutaion call the pair $ (a_i,a_j)$ wrong pair if $ i<j$ and $ a_i >a_j$.Let number of inversion  is number of wrong pair  of permutation $ a_1,a_2,a_3,..,a_n$. Let $ n \ge 2$ is positive integer. Find the number of permutation of $ 1,2,..,n$ such that its  number of inversion is divisible by $ n$.
2008 Mongolia Team Selection Test
Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$  varies on $ \omega$
2012 Morocco TST
Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that :

$$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$
2012 Morocco TST
Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$.

Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.
2012 Morocco TST
$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that :

$$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$
2012 Morocco TST
$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter.

prove that $\widehat{OIH}$ is obtuse.
2012 Morocco TST
Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.
2012 Morocco TST
Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square
2012 Morocco TST
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.



Proposed by Ismail Isaev and Mikhail Isaev, Russia
2011 Morocco TST
Prove that for any n natural, the number $$ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k}  $$

cannot be divided by $5$.
2011 Morocco TST
For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$  passes through a point independent of $ X$.
2011 Morocco TST
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$



Proposed by Nikolay Beluhov, Bulgaria
2010 Morocco TST
$f$ is a function twice differentiable on $[0,1]$ and such that $f''$ is continuous. We suppose that : $f(1)-1=f(0)=f'(1)=f'(0)=0$.

Prove that there exists $x_0$ on $[0,1]$ such that $|f''(x_0)| \geq 4$
2010 Morocco TST
Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that $$ a^2 + b^2 + c^2 \geq \sqrt 3 abc. $$

Cristinel Mortici, Romania
2010 Morocco TST
Any rational number admits a non-decimal representation unlimited decimal expansion. This development has the particularity of being periodic.

Examples: $\frac{1}{7} = 0.142857142857…$ has a period $6$ while $\frac{1}{11}=0.0909090909 …$ $2$ periodic.

What are the reciprocals of the prime integers with a period less than or equal to five?
2010 Morocco TST
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
2010 Morocco TST
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
2010 Morocco TST
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
2010 Morocco TST
Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions:

1) if $f \in G$ and $g \in G$ then $gof \in G$,

2) if $f \in G$ then $f^ {-1} \in G$,

3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$.

Prove  that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$
2005 Morocco TST
Let $A$ be a set of positive integers such that



a) if $a\in A$, the all the positive divisors of $a$ are also in $A$;



b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$.



Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.
2005 Morocco TST
The real numbers $a_1,a_2,...,a_{100}$ satisfy the relationship :

$a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101$

Prove that $|a_k| \leq 10$ for all $k \in \{1,2,...,100\}$
2005 Morocco TST
Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$.

Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively.

Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.
2005 Morocco TST
Find all the functions $f: \mathbb R \rightarrow \mathbb R$ satisfying :

$(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y \in \mathbb R$
2005 Morocco TST
Let $a,b,c$ be positive real numbers. Prove the inequality

$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.$$

When does equality occur?
2005 Morocco TST
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
2005 Morocco TST
A convex quadrilateral  $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent  corners. Show that at least two circles have the same size.
2005 Morocco TST
Find all the positive primes $p$ for which there exist integers $m,n$ satisfying :

$p=m^2+n^2$ and $m^3+n^3-4$ is divisible by $p$.
2005 Morocco TST
Consider the set $A=\{1,2,...,49\}$. We partitionate $A$ into three subsets. Prove that there exist a set from these subsets containing three distincts elements $a,b,c$ such that $a+b=c$
2005 Morocco TST
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that $$\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. $$ Prove that $c\geq1$.
2005 Morocco TST
Let $ABCD$ be a cyclic qudrilaterlal such that $AB.BC=2.CD.DA$

Prove that $8.BD^2 \leq 9.AC^2$
2017 Morocco TST-
Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that;

$$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$
2017 Morocco TST-
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.



Proposed by Evan Chen, Taiwan
2017 Morocco TST-
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP = EQ$.
2019 Nepal TST
Prove that there exist infinitely many pairs of different positive integers $(m, n)$ for which $m!n!$ is a square of an integer.



Proposed by Anton Trygub
2019 Nepal TST
Let $H$ be orthocenter of an acute $\Delta ABC$, $M$ is a midpoint of $AC$. Line $MH$ meets lines $AB, BC$ at points $A_1, C_1$ respectively, $A_2$ and $C_2$ are projections of $A_1, C_1$ onto line $BH$ respectively. Prove that lines $CA_2, AC_2$ meet at circumscribed circle of $\Delta ABC$.



Proposed by Anton Trygub
2019 Nepal TST
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality

$$f(xf(y)) + f(xy) = 2f(x)y$$Proposed by Arseniy Nikolaev
2013 North Korea Team Selection Test
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
2013 North Korea Team Selection Test
Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let  $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows:

$$ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}$$

where

$$ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . $$

Prove that the number of elements in $ X $ is a power of 2.
2013 North Korea Team Selection Test
Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.
2013 North Korea Team Selection Test
Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation.

(1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?

$$ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )$$

$$ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )$$

(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.
2013 North Korea Team Selection Test
The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $  X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.
2013 North Korea Team Selection Test
Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.
2017 Pakistan TST
Let $ABCD$ be a cyclic quadrilateral. The diagonals $AC$ and $BD$ meet at $P$, and $DA $ and $CB$ meet at $Q$. Suppose $PQ$ is perpendicular to $AC$. Let $E$ be the midpoint of $AB$. Prove that $PE$ is perpendicular to $BC$.
2017 Pakistan TST
There are $n$ students in a circle, one behind the other, all facing clockwise. The students have heights $h_1 <h_2 < h_3 < \cdots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or lesss, the two students are permitted to switch places Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.
2017 Pakistan TST
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$



$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
2006 Team Selection Test For CSMO
Find all the pairs of positive numbers such that the last

digit of their sum is 3, their difference is a primer number and

their product is a perfect square.
2006 Team Selection Test For CSMO
Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).
2006 Team Selection Test For CSMO
The set $M= \{1;2;3;\ldots ; 29;30\}$ is  divided in $k$

subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an

integer number $)$, then $a$ and $b$ belong different subsets.

Determine the minimum value of $k$.
2006 Team Selection Test For CSMO
All the squares of a board of $(n+1)\times(n-1)$ squares

are painted with three  colors such that, for any two different

columns and any two different rows, the 4 squares in their

intersections they don't have all the same color. Find the

greatest possible value of $n$.
2019 Peru IMO TST
In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that:



There are no two numbers written in the first row that are equal to each other.

The numbers written in the second row coincide with (in some another order) the numbers written in the first row.

The two numbers written in each column are different and they add up to a rational number.



Determine the maximum quantity of irrational numbers that can be in the chessboard.