Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Saudi Arabia BMO TST
Let $ a > b > c > d $ be positive integers such that

\begin{align*}

a^2 + ac - c^2 = b^2 + bd - d^2

\end{align*}Prove that $ ab + cd $ is a composite number.
2016 Saudi Arabia BMO TST
Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.
2016 Saudi Arabia BMO TST
Does there exist a polynomial $P(x)$ with integral coefficients such that

a)  $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ?

b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?
2016 Saudi Arabia BMO TST
On a chessboard $5 \times 9$ squares, the following game is played.

Initially, a number of frogs are randomly placed on some of the squares, no square containing more than one frog. A turn consists of moving all of the frogs subject to the following rules:

$\bullet$  Each frog may be moved one square up, down, left, or right;

$\bullet$ If a frog moves up or down on one turn, it must move left or right on the next turn, and vice versa;

$\bullet$ At the end of each turn, no square can contain two or more frogs.

The game stops if it becomes impossible to complete another turn. Prove that if initially $33$ frogs are placed on the board, the game must eventually stop. Prove also that it is possible to place $32$ frogs on the board so that the game can continue forever.
2016 Saudi Arabia BMO TST
Given a polynomial $P(x) = a_nx^n + a_{n-1}x^{n-1} + ...+ a_1x + a_0$ of real coefficients. Suppose that $P(x)$ has $n$ real roots (not necessarily distinct), and there exists a positive integer $k$ such that $a_k = a_{k-1} = 0$. Prove that $P(x)$ has a real root of multiplicity $k + 1$.
2016 Saudi Arabia BMO TST
Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$
2016 Saudi Arabia BMO TST
For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.
2016 Saudi Arabia BMO TST
There are There are $64$ towns in a country, and some pairs of towns are connected by roads but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions. but we do not know these pairs. We may choose any pair of towns and find out whether they are connected by a road. Our aim is to determine whether it is possible to travel between any two towns using roads. Prove that there is no algorithm which would enable us to do this in less than $2016$ questions.
2016 Saudi Arabia BMO TST
Given two non-constant polynomials $P(x),Q(x)$ with real coefficients. For a real number $a$, we define

$$P_a= \{z \in C : P(z) = a\}, Q_a =\{z \in C : Q(z) = a\}$$Denote by $K$ the set of real numbers $a$ such that $P_a = Q_a$.

Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.
2016 Saudi Arabia BMO TST
Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA  \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.
2016 Saudi Arabia BMO TST
Let $d$ be a positive integer. Show that for every integer $S$, there exist a positive integer $n$ and a sequence $a_1, ..., a_n \in \{-1, 1\}$ such that  $S = a_1(1 + d)^2 + a_2(1 + 2d)^2 + ... + a_n(1 + nd)^2$.
2016 Saudi Arabia BMO TST
Find all natural numbers $n\geq 3$ satisfying one can cut a convex $n$-gon into different triangles along some of the diagonals (None of these diagonals intersects others at any point other than vertices) and the number of diagonals are used at each vertex is even.
2016 Saudi Arabia BMO TST
Let $ p $ and $ q $ be given primes and the sequence $ \{ p_n \}_{n = 1}^{\infty} $ defined recursively as follows:

$ p_1 = p $, $ p_2 = q $, and $ p_{n+2} $ is the largest prime divisor of the number $( p_n + p_{n + 1} + 2016) $ for all $ n \geq 1 $. Prove that this sequence is bounded. That is, there exists a positive real number $ M $ such that $ p_n < M $ for all positive integers $ n $.
2016 Saudi Arabia BMO TST
Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp  BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly.

a)  Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$.

b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.
2016 Saudi Arabia BMO TST
Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying

$$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$
2016 Saudi Arabia BMO TST
Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.
2015 Saudi Arabia BMO TST
Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$.



Nguyễn Duy Thái Sơn
2015 Saudi Arabia BMO TST
Given $2015$ subsets $A_1, A_2,...,A_{2015}$ of the set $\{1, 2,..., 1000\}$ such that $|A_i| \ge 2$ for every $i \ge 1$ and $|A_i \cap A_j| \ge 1$ for every $1 \le i < j \le 2015$. Prove that $k = 3$ is the smallest number of colors such that we can always color the elements of the set $\{1, 2,..., 1000\}$ by $k$ colors with the property that the subset $A_i$ has at least two elements of different colors for every $i \ge 1$.



Lê Anh Vinh
2015 Saudi Arabia BMO TST
Let $ABC$ be a triangle, $\Gamma$ its circumcircle, $I$ its incenter, and $\omega$ a tangent circle to the line $AI$ at $I$ and to the side $BC$. Prove that the circles $\Gamma$ and $\omega$ are tangent.



Malik Talbi
2015 Saudi Arabia BMO TST
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.



Malik Talbi
2015 Saudi Arabia BMO TST
Find all strictly increasing functions $f : Z \to  R$ such that for any $m, n \in Z$ there exists a $k  \in Z$ such that $f(k) = f(m) - f(n)$.



Nguyễn Duy Thái Sơn
2015 Saudi Arabia BMO TST
Find the number of $6$-tuples $(a_1,a_2, a_3,a_4, a_5,a_6)$ of distinct positive integers satisfying the following two conditions:

(a) $a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 30$

(b) We can write $a_1,a_2, a_3,a_4, a_5,a_6$ on sides of a hexagon such that after a finite number of time choosing a vertex of the hexagon and adding $1$ to  the two numbers written on two sides adjacent to the vertex, we obtain

a hexagon with equal numbers on its sides.



Lê Anh Vinh
2015 Saudi Arabia BMO TST
Let $ABC$ be a triangle, $H_a, H_b$ and $H_c$ the feet of its altitudes from $A, B$ and $C$, respectively, $T_a, T_b, T_c$ its touchpoints of the incircle with the sides $BC, CA$ and $AB$, respectively. The circumcircles of triangles $AH_bH_c$ and $AT_bT_c$ intersect again at $A'$. The circumcircles of triangles $BH_cH_a$ and $BT_cT_a$ intersect again at $B'$. The circumcircles of triangles $CH_aH_b$ and $CT_aT_b$ intersect again at $C'$. Prove that the points $A',B',C'$ are collinear.



Malik Talbi
2015 Saudi Arabia BMO TST
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$.



Lê Anh Vinh
2014 Saudi Arabia BMO TST
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
2014 Saudi Arabia BMO TST
Prove that among any $16$ perfect cubes we can always find two cubes whose difference is divisible by $91$.
2014 Saudi Arabia BMO TST
Let $n \ge 2$ be a positive integer, and write in a digit form $$\frac{1}{n}=0.a_1a_2\dots.$$ Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.
2014 Saudi Arabia BMO TST
Let $ABC$ be a triangle with $\angle B \le \angle C$, $I$ its incenter and $D$ the intersection point of line $AI$ with side $BC$. Let $M$ and $N$ be points on sides $BA$ and $CA$, respectively, such that $BM = BD$ and $CN = CD$. The circumcircle of triangle $CMN$ intersects again line $BC$ at $P$. Prove that quadrilateral $DIMP$ is cyclic.
2014 Saudi Arabia BMO TST
Find all positive integers $n$ such that $$3^n+4^n+\cdots+(n+2)^n=(n+3)^n.$$
2014 Saudi Arabia BMO TST
Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers
2014 Saudi Arabia BMO TST
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.
2014 Saudi Arabia BMO TST
Let $ABCD$ be a parallelogram. A line $\ell$ intersects lines $AB,~ BC,~ CD, ~DA$ at four different points $E,~ F,~ G,~ H,$ respectively. The circumcircles of triangles $AEF$ and $AGH$ intersect again at $P$. The circumcircles of triangles $CEF$ and $CGH$ intersect again at $Q$. Prove that the line $P Q$ bisects the diagonal $BD$.
2014 Saudi Arabia BMO TST
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. (A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex )
2014 Saudi Arabia BMO TST
Let $n > 3$ be an odd positive integer not divisible by $3$. Determine if it is possible to form an $n \times n$ array of numbers such that





(a) the set of the numbers in each row is a permutation of $0, 1, \dots , n - 1$;

the set of the numbers in each column is a permutation of $0, 1, \dots , n-1$;



(b) the board is totally non-symmetric: for $1 \le i < j \le n$ and $1 \le i' < j' \le n$, if $(i, j) \neq (i', j')$ then $(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})$ where $a_{i,j}$ denotes the entry in the $i^\text{th}$ row and $j^\text{th}$ column.
2014 Saudi Arabia BMO TST
Find all functions $f:\mathbb{N}\rightarrow(0,\infty)$ such that $f(4)=4$ and $$\frac{1}{f(1)f(2)}+\frac{1}{f(2)f(3)}+\cdots+\frac{1}{f(n)f(n+1)}=\frac{f(n)}{f(n+1)},~\forall n\in\mathbb{N},$$ where $\mathbb{N}=\{1,2,\dots\}$ is the set of positive integers.
2014 Saudi Arabia BMO TST
Circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$. Segments $AC$ and $BD$ are chords of $\omega_1$ and $\omega_2$ respectively, such that segment $AB$ and ray $CD$ meet at $P$. Ray $BD$ and segment $AC$ meet at $X$. Point $Y$ lies on $\omega_1$ such that $P Y \parallel BD$. Point $Z$ lies on $\omega_2$ such that $P Z \parallel AC$. Prove that points $Q,~ X,~ Y,~ Z$ are collinear.
2014 Saudi Arabia BMO TST
Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that $$\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.$$
2014 Saudi Arabia BMO TST
Let $f :\mathbb{N} \rightarrow\mathbb{N}$ be an injective function such that $f(1) = 2,~ f(2) = 4$ and $$f(f(m) + f(n)) = f(f(m)) + f(n)$$ for all $m, n \in \mathbb{N}$. Prove that $f(n) = n + 2$ for all $n \ge 2$.
2014 Saudi Arabia BMO TST
Let $ABC$ be a triangle. Circle $\Omega$ passes through points $B$ and $C$. Circle $\omega$ is tangent internally to $\Omega$ and also to sides $AB$ and $AC$ at $T,~ P,$ and $Q$, respectively. Let $M$ be midpoint of arc $\widehat{BC}$ (containing T) of $\Omega$. Prove that lines $P Q,~ BC,$ and $MT$ are concurrent.
2013 Saudi Arabia BMO TST
The set $G$ is defined by the points $(x,y)$ with integer coordinates, $1 \le x \le  5$ and $1 \le  y \le 5$. Determine the number of five-point sequences $(P_1, P_2, P_3, P_4, P_5)$ such that for $1 \le  i \le 5$, $P_i = (x_i,i)$ is in $G$ and $|x_1 - x_2| = |x_2 - x_3| = |x_3 -  x_4|=|x_4 - x_5| = 1$.
2013 Saudi Arabia BMO TST
For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$
2013 Saudi Arabia BMO TST
Solve the following equation where $x$ is a real number: $\lfloor x^2 \rfloor  -10\lfloor x \rfloor  + 24 = 0$
2013 Saudi Arabia BMO TST
$ABCDEF$ is an equiangular hexagon of perimeter $21$. Given that $AB = 3, CD = 4$, and $EF = 5$, compute the area of hexagon $ABCDEF$.
2013 Saudi Arabia BMO TST
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$is $-2013$. Find the sum of the squares of these real roots.
2013 Saudi Arabia BMO TST
Let $ABC$ be a triangle with incenter $I,$ and let $D,E,F$ be the midpoints of sides $BC, CA, AB$, respectively. Lines $BI$ and $DE$ meet at $P $ and lines $CI$ and $DF$ meet at $Q$. Line $PQ$ meets sides $AB$ and $AC$ at $T$ and $S$, respectively. Prove that $AS = AT$
2013 Saudi Arabia BMO TST
Ayman wants to color the cells of a $50 \times  50$ chessboard into black and white so that each $2 \times  3$ or $3 \times  2$ rectangle contains an even number of white cells. Determine the number of ways Ayman can color the chessboard.
2013 Saudi Arabia BMO TST
Prove that the ratio $$\frac{1^1 + 3^3 + 5^5 + ...+ (2^{2013} - 1)^{(2^{2013} - 1)}}{2^{2013}}$$is an odd integer.
2013 Saudi Arabia BMO TST
In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.
2013 Saudi Arabia BMO TST
Find all functions $f : R \to  R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$
2013 Saudi Arabia BMO TST
Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$
2013 Saudi Arabia BMO TST
Ten students are standing in a line. A teacher wants to place a hat on each student. He has two colors of hats, red and white, and he has $10$ hats of each color. Determine the number of ways in which the teacher can place hats such that among any set of consecutive students, the number of students with red hats and the number of students with blue hats differ by at most $2$
2013 Saudi Arabia BMO TST
We call a positive integer good if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.
2013 Saudi Arabia BMO TST
Let $a, b,c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that

$$a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3$$
2013 Saudi Arabia BMO TST
The excircle $\omega_B$ of triangle $ABC$ opposite $B$ touches side $AC$, rays $BA$ and $BC$ at $B_1, C_1$ and $A_1$, respectively. Point $D$ lies on major arc $A_1C_1$ of $\omega_B$. Rays $DA_1$ and $C_1B_1$ meet at $E$. Lines $AB_1$ and $BE$ meet at $F$. Prove that line $FD$ is tangent to $\omega_B$ (at $D$).
2013 Saudi Arabia BMO TST
A social club has $101$ members, each of whom is fluent in the same $50$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of $A$.
2013 Saudi Arabia BMO TST
$ABCD$ is a cyclic quadrilateral and $\omega$ its circumcircle. The perpendicular line to $AC$ at $D$ intersects $AC$ at $E$ and $\omega$ at F. Denote by $\ell$ the perpendicular line to $BC$ at $F$. The perpendicular line to $\ell$ at A intersects $\ell$ at $G$ and $\omega$ at $H$. Line $GE$ intersects $FH$ at $I$ and $CD$ at $J$. Prove that points $C, F, I$ and $J$ are concyclic
2013 Saudi Arabia BMO TST
Define Fibonacci sequence $\{F\}_{n=0}^{\infty}$ as $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n +F_{n-1}$ for every integer $n > 1$. Determine all quadruples $(a, b, c,n)$ of positive integers with a $< b < c$ such that each of $a, b,c,a + n, b + n,c + 2n$ is a term of the Fibonacci sequence.
2013 Saudi Arabia BMO TST
Let $T$ be a real number satisfying the property:

For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$.

Determine the minimum value of $T$.
2013 Saudi Arabia BMO TST
Let $f : Z_{\ge 0} \to  Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$:

(a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$,  (b) $f(2n) \ge  f(n)$

where $Z_{\ge 0}$  is the set of nonnegative integers. Solve the equation $f(n) = 1000$
2013 Saudi Arabia BMO TST
$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.
2013 Saudi Arabia BMO TST
The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$?
2013 Saudi Arabia BMO TST
Find the area of the set of points of the plane whose coordinates $(x, y)$ satisfy $x^2 + y^2 \le  4|x| + 4|y|$.
2013 Saudi Arabia BMO TST
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.
2019 Saudi Arabia IMO TST
Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le  f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers
2019 Saudi Arabia IMO TST
Find all pair of integers $(m,n)$ and $m \ge  n$ such that there exist a positive integer $s$ and

a) Product of all divisor of $sm, sn$ are equal.

b) Number of divisors of $sm,sn$ are equal.
2019 Saudi Arabia IMO TST
Let regular hexagon is divided into $6n^2$ regular triangles. Let $2n$ coins are put in different triangles such, that no any two coins lie on the same layer (layer is area between two consecutive parallel lines). Let also triangles are painted like on the chess board. Prove that exactly $n$ coins lie on black triangles.
2019 Saudi Arabia IMO TST
Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$
2019 Saudi Arabia IMO TST
Let non-constant polynomial $f(x)$ with real coefficients is given with the following property:

for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in  Z$$Prove that $f(x)$ is divisible by $x$
2019 Saudi Arabia IMO TST
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$
2018 Saudi Arabia IMO TST
Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that

i. All terms of sequences are pairwise coprime.

ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded.

Prove that this sequence contains infinitely many primes.
2018 Saudi Arabia IMO TST
Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following:

i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order);

ii. The sum of all numbers in each row is $n$.

Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold.

Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.
2018 Saudi Arabia IMO TST
Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.
2018 Saudi Arabia IMO TST
Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of  triangle $DEF$ is the midpoint of segment $IH$.
2018 Saudi Arabia IMO TST
a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with  $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$.

b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.
2018 Saudi Arabia IMO TST
Two sets of positive integers $A, B$ are called connected  if they are not empty and for all $a  \in  A, b \in B$, number $ab + 1$ is a perfect square.

i)  Given $A =\{1, 2,3, 4\}$. Prove that there does not exist any set $B$ such that $A, B$ are connected.

ii)  Suppose that $A, B$ are connected with  $|A|,|B| \ge 2$. For any $a_1 > a_2 \in A$ and $b_1 > b_2 \in B$, prove that $a_1b_1 > 13a_2b_2$.
2018 Saudi Arabia IMO TST
Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$.

Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).
2018 Saudi Arabia IMO TST
A non-empty subset of $\{1,2, ..., n\}$ is called arabic  if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of  $\{1,2, ..., n\}$  has the same parity as $n$.
2018 Saudi Arabia IMO TST
Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.
2018 Saudi Arabia IMO TST
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne  4$, and

max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$  for all $m, n \in Z^+$.