Navvye/Olympiad-Problems-CSV huggingface dataset:
1978 Romania Team Selection Test
Prove that for every partition of $ \{ 1,2,3,4,5,6,7,8,9\} $ into two subsets, one of the subsets contains three numbers such that the sum of two of them is equal to the double of the third.
1978 Romania Team Selection Test
Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $

Prove that there exists an integer $ n $ such that $ k=11^nl. $
1978 Romania Team Selection Test
Let $ P[X,Y] $ be a polynomial of degree at most $ 2 .$ If $ A,B,C,A',B',C' $ are distinct roots of $ P $ such that $ A,B,C $ are not collinear and  $ A',B',C' $ lie on the lines $ BC,CA, $ respectively, $ AB, $ in the planar representation of these points, show that $ P=0. $
1978 Romania Team Selection Test
Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?
1978 Romania Team Selection Test
Prove that there is no square with its four vertices on four concentric circles whose radii form an arithmetic progression.
1978 Romania Team Selection Test
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.
1978 Romania Team Selection Test
Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?
1978 Romania Team Selection Test
For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are similar, if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $



a) If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them.

b) For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.
1978 Romania Team Selection Test
A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $  x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $

Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.
1978 Romania Team Selection Test
Associate to any point $ (h,k) $ in the integer net of the cartesian plane a real number $ a_{h,k} $ so that

$$ a_{h,k}=\frac{1}{4}\left( a_{h-1,k} +a_{h+1,k}+a_{h,k-1}+a_{h,k+1}\right) ,\quad\forall h,k\in\mathbb{Z} . $$

a) Prove that it´s possible that all the elements of the set $ A:=\left\{ a_{h,k}\big| h,k\in\mathbb{Z}\right\} $ are different.

b) If so, show that the set $ A $ hasn´t any kind of boundary.
1978 Romania Team Selection Test
Prove that there is a function $ F:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ (F\circ F) (n) =n^2, $ for all $ n\in\mathbb{N} . $
1978 Romania Team Selection Test
Let $ A_1,A_2,...,A_{3n} $ be $ 3n\ge 3 $ planar points such that $ A_1A_2A_3 $ is an equilateral triangle and $ A_{3k+1} ,A_{3k+2} ,A_{3k+3} $ are the midpoints of the sides of $ A_{3k-2}A_{3k-1}A_{3k} , $ for all $ 1\le k<n. $ Of two different colors, each one of these points are colored, either with one, either with another.



a) Prove that, if $ n\ge 7, $ then some of these points form a monochromatic (only one color) isosceles trapezoid.

b) What about $ n=6? $
1978 Romania Team Selection Test
Let $ \mathcal{M} $ a set of $ 3n\ge 3 $ planar points such that the maximum distance between two of these points is $ 

1 $. Prove that:



a) among any four points,there are two aparted by a distance at most $ \frac{1}{\sqrt{2}} . $

b) for $ n=2 $ and any $ \epsilon >0, $ it is possible that $ 12 $ or $ 15 $ of the distances between points from $ \mathcal{M} $ lie in the interval $ (1-\epsilon , 1]; $ but any $ 13 $ of the distances can´t be found all in the interval $ \left(\frac{1}{\sqrt 2} ,1\right]. $

c) there exists a circle of diameter $ \sqrt{6} $ that contains $ \mathcal{M} . $

d) some two points of $ \mathcal{M} $ are on a distance not exceeding $ \frac{4}{3\sqrt n-\sqrt 3} . $
1978 Romania Team Selection Test
In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $



a) Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid?

b) The same question with the addition that $ \angle BAD $ is obtuse.
1978 Romania Team Selection Test
Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.
1978 Romania Team Selection Test
a) Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $



b) Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants.



c) Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.
1978 Romania Team Selection Test
Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $
1978 Romania Team Selection Test
Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that

$$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$
1978 Romania Team Selection Test
a) Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $



b) Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $
1978 Romania Team Selection Test
a) Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers).



b) Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.
1978 Romania Team Selection Test
Show that for every natural number $ a\ge 3, $ there are infinitely many natural numbers $ n $ such that $ a^n\equiv 1\pmod n . $ Does this hold for $ n=2? $
1978 Romania Team Selection Test
Let $ k $ be a natural number. A function $ f:S:=\left\{ x_1,x_2,...,x_k\right\}\longrightarrow\mathbb{R} $ is said to be additive if, whenever $ n_1x_1+n_2x_2+\cdots +n_kx_k=0, $ it holds that $ n_1f\left( x_1\right)+n_2f\left( x_2\right)+\cdots +n_kf\left( x_k\right)=0, $ for all natural numbers $ n_1,n_2,...,n_k. $



Show that for every additive function and for every finite set of real numbers $ T, $ there exists a second function, which is a real additive function defined on $ S\cup T $ and which is equal to the former on the restriction $ S. $
1978 Romania Team Selection Test
Let $ p $ be a natural number and let two partitions $ \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} $ of a finite set $ \mathcal{M} . $ Knowing that, whenever an element of $ \mathcal{A} $ doesn´t have any elements in common with another of $ \mathcal{B} , $ it holds that the number of elements of these two is greater than $ p, $ prove that $ \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . $ Can equality hold?
1978 Romania Team Selection Test
Let be some points on a plane, no three collinear. We associate a positive or a negative value to every segment formed by these. Prove that the number of points, the number of segments with negative associated value, and the number of triangles that has a negative product of the values of its sides, share the same parity.
2019 SAFEST Olympiad
Let $ABC$ be an isosceles triangle with $AB = AC$. Let $AD$ be the diameter of the circumcircle of $ABC$ and let $P$ be a point on the smaller arc $BD$. The line $DP$ intersects the rays $AB$ and $AC$ at points $M$ and $N$, respectively. The line $AD$ intersects the lines $BP$ and $CP$ at points $Q$ and $R$, respectively. Prove that the midpoint of $MN$ lies on the circumcircle of $PQR$
2019 SAFEST Olympiad
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$Prove that the total degree of $f$ is at least $n$.
2019 SAFEST Olympiad
Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$.



Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$  this minimum occurs
2019 SAFEST Olympiad
There are $25$ IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least $10$ other participants
2019 SAFEST Olympiad
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2019 Saudi Arabia BMO TST
Let $p$ be an odd prime number.

a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.

b) Find all $n$ satisfy condition above when $p = 3$
2019 Saudi Arabia BMO TST
Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic
2019 Saudi Arabia BMO TST
For $n \ge  3$, it is given an $2n \times 2n$ board with black and white squares. It is known that all border squares are black and no $2 \times  2$ subboard has all four squares of the same color. Prove that there exists a $2 \times  2$ subboard painted like a chessboard, i.e. with two opposite black corners and two opposite white corners.
2019 Saudi Arabia BMO TST
There are $n$ people with hats present at a party. Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment. Find all $n \ge  2$ for which the order of

greetings can be arranged in such a way that after all of them, each person has their own hat back.
2019 Saudi Arabia BMO TST
Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$  and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $
2019 Saudi Arabia BMO TST
The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.
2019 Saudi Arabia BMO TST
Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.
2019 Saudi Arabia BMO TST
Let $ABCD$ is a trapezoid with $\angle A = \angle B = 90^o$ and let $E$ is a point lying on side $CD$. Let the circle $\omega$ is inscribed to triangle $ABE$ and tangents sides $AB, AE$ and $BE$ at points $P, F$ and $K$ respectively. Let $KF$ intersects segments $BC$ and $AD$ at points $M$ and $N$ respectively, as well as $PM$ and $PN$ intersect $\omega$ at points $H$ and $T$ respectively. Prove that $PH = PT$.
2019 Saudi Arabia BMO TST
Let $300$ students participate to the Olympiad. Between each $3$ participants there is a pair that are not friends. Hamza enumerates participants in some order and denotes by $x_i$ the number of friends of $i$-th participant. It occurs that $\{x_1,x_2,...,x_{299},x_{300}\} = \{1, 2,..., N - 1,N\}$ Find the biggest possible value for $N$.
2018 Saudi Arabia BMO TST
Find the smallest positive integer $n$ which can not be expressed as $n =\frac{2^a - 2^b}{2^c - 2^d}$ for some positive integers $a, b, c, d$
2018 Saudi Arabia BMO TST
Find all functions $f : R \to R$ such that  $f( 2x^3 + f (y)) = y + 2x^2 f (x)$  for all real numbers $x, y$.
2018 Saudi Arabia BMO TST
The partition of $2n$ positive integers into $n$ pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for $2n$ distinct positive integers, there exists one square-free partition, then there exists at least $n!$ square-free partitions.
2018 Saudi Arabia BMO TST
Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \perp BE$ and $AN \perp  C D$ respectively.

a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.

b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
2018 Saudi Arabia BMO TST
Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$.

a) Prove that $DE$ is parallel to $BC$.

b)  Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.
2018 Saudi Arabia BMO TST
Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.
2018 Saudi Arabia BMO TST
Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.
2018 Saudi Arabia BMO TST
Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ ,

for all $x, y \in Z$, $x \ne 0$.
2017 Saudi Arabia BMO TST
Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called nice  if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)
2017 Saudi Arabia BMO TST
Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.
2017 Saudi Arabia BMO TST
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?
2017 Saudi Arabia BMO TST
Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$.

a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number.

b) A positive integer is called Fib-unique if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.
2017 Saudi Arabia BMO TST
Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers
2017 Saudi Arabia BMO TST
Polynomial P(x) with integer coefficient is called cube-presented if it can be represented as sum of several cube of polynomials with integer coefficients.

Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$.

a) Is $3x^2$ a cube-represented polynomial?

b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?
2017 Saudi Arabia BMO TST
We put four numbers $1,2, 3,4$ around a circle in order. One starts at the number $1$ and every step, he moves to an adjacent number on either side. How many ways he can move such that sum of the numbers he visits in his path (including the starting number) is equal to $21$?
2017 Saudi Arabia BMO TST
Let $ABC$ be a triangle with $A$ is an obtuse angle. Denote $BE$ as the internal angle bisector of triangle $ABC$ with $E \in AC$ and suppose that $\angle AEB = 45^o$. The altitude $AD$ of triangle $ABC$ intersects $BE$ at $F$. Let $O_1, O_2$ be the circumcenter of triangles $FED, EDC$. Suppose that $EO_1, EO_2$ meet $BC$ at $G, H$ respectively. Prove that $\frac{GH}{GB}= \tan \frac{a}{2}$
2017 Saudi Arabia BMO TST
Prove that there are infinitely many positive integer $n$ such that $n!$  is divisible by $n^3 -1$.
2017 Saudi Arabia BMO TST
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied:

i) $2f (x) + 2f (y) \le  f (x + y)$

ii) $(x + y)[y f (x) + x f (y)]  \ge x y f (x + y)$
2017 Saudi Arabia BMO TST
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.

a)  Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.

b)  Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
2017 Saudi Arabia BMO TST
Consider the set $X =\{1, 2,3, ...,2018\}$.

How many positive integers $k$ with $2 \le  k  \le 2017$ that satisfy the following conditions:

i) There exists some partition of the set $X$ into $1009$ disjoint pairs which are $(a_1, b_1),(a_2, b_2), ...,(a_{1009}, b_{1009})$ with $|a_i - b_i| \in \{1, k\}$.

ii) For all partitions satisfy the condition (i), the sum $T = \sum^{1009}_{i=1} |a_i - b_i|$ has the right most digit is $9$
2017 Saudi Arabia BMO TST
Let $a, b, c$ be positive real numbers. Prove that

$$\frac{a(b^2 + c^2)}{(b + c)(a^2 + bc)} + \frac{b(c^2 + a^2)}{(c + a)(b^2 + ca)} + \frac{c(a^2 + b^2)}{(a + b)(c^2 + ab)} \ge  \frac32$$
2017 Saudi Arabia BMO TST
Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$
2017 Saudi Arabia BMO TST
Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$.

a) Prove that $SF \perp EF.$

b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$
2017 Saudi Arabia BMO TST
Let $p$ be a prime number and a table of size $(p^2+ p+1)\times (p^2+p + 1)$ which is divided into unit cells. The way to color some cells of this table is called nice if there are no four colored cells that form a rectangle (the sides of rectangle are parallel to the sides of given table).

1. Let $k$ be the number of colored cells in some nice coloring way. Prove that $k \le  (p + 1)(p^2 + p + 1)$. Denote this number as $k_{max}$.

2. Prove that all ordered tuples $(a, b, c)$ with $0 \le a, b, c < p$ and $a + b + c > 0$ can be partitioned into $p^2 + p + 1$ sets $S_1, S_2, .. . S_{p^2+p+1}$ such that two tuples $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ belong to the same set if and only if $a_1 \equiv ka_2, b_1 \equiv kb_2, c_1 \equiv kc_2$ (mod $p$) for some $k \in \{1,2, 3, ... , p - 1\}$.

3. For $1 \le i, j \le  p^2+p+1$, if there exist $(a_1, b_1, c_1) \in S_i$ and $(a_2, b_2, c_2) \in S_j$ such that $a_1a_2 + b_1b_2 + c_1c_2 \equiv  0$ (mod $p$), we color the cell $(i, j)$ of the given table. Prove that this coloring way is nice with $k_{max}$ colored cells
2016 Saudi Arabia BMO TST
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.
2016 Saudi Arabia BMO TST
Let $A$ be a point outside the circle $\omega$. Two points $B, C$ lie on $\omega$ such that $AB, AC$ are tangent to $\omega$. Let $D$ be any point on $\omega$ ($D$ is neither $B$ nor $C$) and $M$ the foot of perpendicular from $B$ to $CD$. The line through $D$ and the midpoint of $BM$ meets $\omega$ again at $P$. Prove that $AP \perp CP$
2016 Saudi Arabia BMO TST
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
2016 Saudi Arabia BMO TST
On a checkered square $10 \times  10$ the cells of the upper left $5  \times 5$ square are black and all the other cells are white. What is the maximal $n$ such that the original square can be dissected (along the borders of the cells) into $n$ polygons such that in each of them the number of black cells is three times less than the number of white cells? (The polygons need not be congruent or even equal in area.)
2016 Saudi Arabia BMO TST
Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le  i < j \le n ,  i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root.  Find the greatest possible value of $n$.
2016 Saudi Arabia BMO TST
A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.
2016 Saudi Arabia BMO TST
Let $ m $ and $ n $ be odd integers such that $n^2 - 1 $ is divisible by $m^2 + 1 - n^2 $.

Prove that $ |m^2 + 1 - n^2 | $ is a perfect square.
2016 Saudi Arabia BMO TST
How many ways are there to color the vertices of a square with $n$ colors $1,2, .. ., n$. (The colorings must be different so that we can’t get one from the other by a rotation.)