Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 Saudi Arabia IMO TST
Let $ABC$ be an acute-angled triangle inscribed in circle $(O)$. Let $G$ be a point on the small arc $AC$ of $(O)$ and $(K)$ be a circle passing through $A$ and $G$. Bisector of $\angle BAC$ cuts $(K)$ again at $P$. The point $E$ is chosen on $(K)$ such that $AE$ is parallel to $BC$. The line $PK$ meets the perpendicular bisector of $BC$ at $F$. Prove that $\angle EGF = 90^o$.
2018 Saudi Arabia IMO TST
Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy  $|a_i - i| = k$ for all $i = 1,1000$.
2017 Saudi Arabia IMO TST
Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.
2017 Saudi Arabia IMO TST
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying:

$$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$
2017 Saudi Arabia IMO TST
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.
2017 Saudi Arabia IMO TST
In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?
2017 Saudi Arabia IMO TST
Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.
2017 Saudi Arabia IMO TST
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
2017 Saudi Arabia IMO TST
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n  \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
2017 Saudi Arabia IMO TST
Let $ABCD$ be a quadrilateral inscribed a circle $(O)$. Assume that $AB$ and $CD$ intersect at $E, AC$ and $BD$ intersect at $K$, and $O$ does not belong to the line $KE$. Let $G$ and $H$ be the midpoints of $AB$ and $CD$ respectively. Let $(I)$ be the circumcircle of the triangle $GKH$. Let $(I)$ and $(O)$ intersect at $M, N$ such that $MGHN$ is convex quadrilateral. Let $P$ be the intersection of $MG$ and $HN,Q$ be the intersection of $MN$ and $GH$.

a) Prove that $IK$ and $OE$ are parallel.

b) Prove that $PK$ is perpendicular to $IQ$.
2017 Saudi Arabia IMO TST
The $64$ cells of an $8 \times  8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?
2017 Saudi Arabia IMO TST
Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\}  \ge 1$. Prove that

$$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)}  \le (\frac{a+b+c}{3} )^2 + 1 $$
2017 Saudi Arabia IMO TST
Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that

a)  For every integer $n$, we have  $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$

b)  The value $\frac{1}{2\sqrt{17} }$  is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$
2017 Saudi Arabia IMO TST
For integer $n > 1$, consider $n$ cube polynomials $P_1(x), ..., P_n(x)$ such that each polynomial has $3$ distinct real roots. Denote $S$ as the set of roots of following equation $P_1(x)P_2(x)P_3(x)... P_n(x) = 0$.

It is also known that for each $1 \le i < j \le  n, P_i(x)P_j(x) = 0$ has $5$ distinct real roots.

1. Prove that if for each $a, b \in S$, there is exactly one $i \in\{1,2, 3,..., n\}$ such that $P_i(a) = P_i(b) = 0$ then $n = 7$.

2. Prove that if $n > 7$ then $|S| = 2n + 1$.
2016 Saudi Arabia IMO TST
Let $ n \geq 3 $ be an integer and let

\begin{align*}

x_1,x_2, \ldots, x_n

\end{align*}be $ n $ distinct integers. Prove that

\begin{align*}

(x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6.

\end{align*}
2016 Saudi Arabia IMO TST
Given a set of $2^{2016}$ cards with the numbers $1,2, ..., 2^{2016}$ written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the $2^{2015}$ remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have $2^{2014}$ cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.
2016 Saudi Arabia IMO TST
Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.
2016 Saudi Arabia IMO TST
Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that

\begin{align*}

x^2 + 6y^2 = 7^k.

\end{align*}
2016 Saudi Arabia IMO TST
Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.
2016 Saudi Arabia IMO TST
Find all functions $f : R \to  R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.
2016 Saudi Arabia IMO TST
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le  k  \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
2016 Saudi Arabia IMO TST
Find all pairs of polynomials $P(x),Q(x)$ with integer coefficients such that $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$ for all real numbers $x$
2016 Saudi Arabia IMO TST
Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that:

$\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot  2016$.

$\bullet$  The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$.

Find the maximum value of $n$
2016 Saudi Arabia IMO TST
Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$.

a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$.

b) Prove that $A, K, L$ are collinear.
2016 Saudi Arabia IMO TST
Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq  x,  y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.
2016 Saudi Arabia IMO TST
Find the number of permutations $ ( a_1, a_2, .  \ . \ , a_{2016}) $ of the first $ 2016 $ positive integers satisfying the following two

conditions:



1. $ a_{i+1} - a_i \leq 1$ for all $i = 1, 2, . \ . \ . , 2015$, and

2. There are exactly two indices $ i < j $ with $ 1 \leq i < j \leq 2016 $ such that $ a_i = i $

and $  a_j = j$.
2016 Saudi Arabia IMO TST
Call a positive integer $N \ge 2$ special  if for every k such that $2 \le k \le  N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). Find all special positive integers.
2016 Saudi Arabia IMO TST
Let $ABC$ be a triangle inscribed in $(O)$. Two tangents of $(O)$ at $B,C$ meets at $P$. The bisector of angle $BAC $ intersects $(P,PB)$ at point $E$ lying inside triangle $ABC$. Let $M,N$ be the midpoints of arcs $BC$ and $BAC$. Circle with diameter $BC$ intersects line segment $EN$ at $F$. Prove that the orthocenter of triangle $EFM$ lies on $BC$.
2016 Saudi Arabia IMO TST
Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.
2016 Saudi Arabia IMO TST
Find all functions $f : R \to  R$ satisfying the conditions:

1. $f (x + 1) \ge  f (x) + 1$ for all $x \in R$

2. $f (x y) \ge f (x)f (y)$ for all $x, y \in R$
2016 Saudi Arabia IMO TST
Given two positive integers $r > s$, and let $F$ be an infinite family of sets, each of size $r$, no two of which share fewer than $s$ elements. Prove that there exists a set of size $r -1$ that shares at least $s$ elements with each set in $F$.
2016 Saudi Arabia IMO TST
Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is nice  if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,
2016 Saudi Arabia IMO TST
On the Cartesian coordinate system $Oxy$, consider a sequence of points $A_n(x_n, y_n)$ in which $(x_n)^{\infty}_{n=1}$,$(y_n)^{\infty}_{n=1}$ are two sequences of positive numbers satisfing the following conditions:

$$x_{n+1} =\sqrt{\frac{x_n^2+x_{n+2}^2}{2}}, y_{n+1} =\big( \frac{\sqrt{y_n}+\sqrt{y_{n+2}}}{2} \big)^2  \,\, \forall n \ge   1 $$Suppose that $O, A_1, A_{2016}$ belong to a line $d$ and $A_1, A_{2016}$ are distinct.

Prove that all the points $A_2, A_3,. .. , A_{2015}$ lie on one side of $d$.
2016 Saudi Arabia IMO TST
Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle  B = \angle  C +\angle  D = \angle  E +\angle  F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$.



Tran Quang Hung
2015 Saudi Arabia IMO TST
Find all functions $f : R_{>0} \to  R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers.



Nguyễn Duy Thái Sơn
2015 Saudi Arabia IMO TST
Let $ABC$ be a triangle with orthocenter $H$. Let $P$ be any point of the plane of the triangle. Let $\Omega$ be the circle with the diameter $AP$ . The circle $\Omega$  cuts $CA$ and $AB$ again at $E$ and $F$ , respectively. The line $PH$ cuts $\Omega$  again at $G$. The tangent lines to $\Omega$  at $E, F$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $L$ be the point on $MG$ such that $AL$ and $MT$ are parallel. Prove that $LA$ and $LH$ are orthogonal.



Lê Phúc Lữ
2015 Saudi Arabia IMO TST
Let $n$ and $k$ be two positive integers. Prove that if $n$ is relatively prime with $30$, then there exist two integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2 - b^2 + k}{n}$ is an integer.



Malik Talbi
2015 Saudi Arabia IMO TST
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$.



Lê Phúc Lữ
2015 Saudi Arabia IMO TST
Hamza and Majid play a game on a horizontal $3 \times 2015$ white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal $1 \times 3$ rectangle. A legal move for Majid consists of painting three unit squares forming a vertical $3\times 1$ rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win?



Lê Anh Vinh
2015 Saudi Arabia IMO TST
Let $a_1, a_2, ...,a_n$ be positive real numbers such that $$a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$$Prove that $$\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge  0$$Võ Quốc Bá Cẩn.
2015 Saudi Arabia IMO TST
Let $S$ be a positive integer divisible by all the integers $1, 2,...,2015$ and $a_1, a_2,..., a_k$ numbers in $\{1, 2,...,2015\}$ such that $2S \le a_1 + a_2 + ... + a_k$. Prove that we can select from $a_1, a_2,..., a_k$ some numbers so that the sum of these selected numbers is equal to $S$.



Lê Anh Vinh
2015 Saudi Arabia IMO TST
Let $ABC$ be a triangle and $\omega$ its circumcircle. Point $D$ lies on the arc $BC$ (not containing $A$) of $\omega$ and is different from $B, C$ and the midpoint of arc $BC$ . The tangent line to $\omega$ at $D$ intersects lines $BC, CA,AB$ at $A', B',C'$ respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA' $ intersects again circle $\omega$ at $F$. Prove that the three points $D,E,F$ are colinear.



Malik Talbi
2015 Saudi Arabia IMO TST
Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have

• The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$,

• If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$,

• If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$.



Lê Anh Vinh
2015 Saudi Arabia IMO TST
Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$.



Trần Nam Dũng
2015 Saudi Arabia IMO TST
The total number of languages used in KAUST is $n$. For each positive integer $k \le n$, let $A_k$ be the set of all those people in KAUST who can speak at least $k$ languages; and let $B_k$ be the set of all people $P$ in KAUST with the property that, for any $k$ pairwise different languages (used in KAUST), $P$ can speak at least one of these $k$ languages. Prove that

(a) If $2k \ge  n + 1$ then $A_k  \subseteq B_k$

(b) If $2k \le  n + 1$ then $A_k \supseteq B_k.$



Nguyễn Duy Thái Sơn
2015 Saudi Arabia IMO TST
Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} +  \frac{1}{y} +  \frac{1}{z}\right)= 10$$Find the greatest value and the least value of

$$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} +  \frac{1}{y^2} +  \frac{1}{z^2}\right)$$Trần Nam Dũng
2014 Saudi Arabia IMO TST
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than $100$. Then Sultan is telling a number greater than $1$. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number.  Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?
2014 Saudi Arabia IMO TST
Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not both appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.
2014 Saudi Arabia IMO TST
Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.
2014 Saudi Arabia IMO TST
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(n+1)>\frac{f(n)+f(f(n))}{2}$$ for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.
2014 Saudi Arabia IMO TST
Let $\Gamma$ be a circle with center $O$ and $AE$ be a diameter. Point $D$ lies on segment $OE$ and point $B$ is the midpoint of one of the arcs $\widehat{AE}$ of $\Gamma$. Construct point $C$ such that $ABCD$ is a parallelogram. Lines $EB$ and $CD$ meet at $F$. Line $OF$ meets the minor arc $\widehat{EB}$ at $I$. Prove that $EI$ bisects $\angle BEC$.
2014 Saudi Arabia IMO TST
Let $S$ be a set of positive real numbers with five elements such that for any distinct $a,~ b,~ c$ in $S$, the number $ab + bc + ca$ is rational. Prove that for any $a$ and $b$ in $S$, $\tfrac{a}{b}$ is a rational number.
2014 Saudi Arabia IMO TST
Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.
2014 Saudi Arabia IMO TST
Aws plays a solitaire game on a fifty-two card deck: whenever two cards of the same color are adjacent, he can remove them. Aws wins the game if he removes all the cards. If Aws starts with the cards in a random order, what is the probability for him to win?
2014 Saudi Arabia IMO TST
A perfect number is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number.



(a) A square-free integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers.



(b) Prove that no perfect square is a perfect number.
2014 Saudi Arabia IMO TST
Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and $$f(x)=1+5f\left( \left\lfloor \frac{x}{2} \right \rfloor\right)-6f\left(\left \lfloor\frac{x}{4} \right \rfloor \right)$$for all $x>0$.
2014 Saudi Arabia IMO TST
There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.
2014 Saudi Arabia IMO TST
Let $\omega_1$ and $\omega_2$ with center $O_1$ and $O_2$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. Lines $XA$ and $Y A$ meet $\omega_2$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_1O_2$, $S$ be the midpoint of $XA$ and $T$ be the midpoint of $W A$. Prove that $MS = MT$ if and only if $X,~ Y ,~ Z$ and $W$ are concyclic.
2014 Saudi Arabia IMO TST
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that $$\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.$$ When does equality hold?
2014 Saudi Arabia IMO TST
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?
2014 Saudi Arabia IMO TST
We are given a lattice and two pebbles $A$ and $B$ that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?
2014 Saudi Arabia IMO TST
Points $A_1,~ B_1,~ C_1$ lie on the sides $BC,~ AC$ and $AB$ of a triangle $ABC$, respectively, such that $AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1$. Let $I_A,~ I_B,~ I_C$ be the incenters of triangles $AB_1C_1,~ A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcenter of triangle $I_AI_BI_C$, is the incenter of triangle $ABC$.
2013 Saudi Arabia IMO TST
Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.
2013 Saudi Arabia IMO TST
Let $S = f\{0.1. 2.3,...\}$ be the set of the non-negative integers. Find all strictly increasing functions $f : S \to S$ such that $n + f(f(n)) \le  2f(n)$ for every $n$ in $S$
2013 Saudi Arabia IMO TST
A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that :

(a) each pair of correspondents from the same office share exactly one common correspondent from the other office.

(b) there are at least $10$ correspondents from Riyadh.

(c) Zayd, one of the correspondents from Jeddah, is in contact with exactly $8$ correspondents from Riyadh.

What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?
2013 Saudi Arabia IMO TST
Determine whether it is possible to place the integers $1, 2,...,2012$ in a circle in such a way that the $2012$ products of adjacent pairs of numbers leave pairwise distinct remainders when divided by $2013$.
2013 Saudi Arabia IMO TST
Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
2013 Saudi Arabia IMO TST
Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.