Navvye/Olympiad-Problems-CSV huggingface dataset:
1995 China Team Selection Test
$ A$ and $ B$ play the following game with a polynomial of degree at least 4:

$$ x^{2n} +  \_x^{2n - 1} +  \_x^{2n - 2} + \ldots + \_x + 1 = 0

$$

$ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up.  If the resulting polynomial has no real roots, $ A$ wins.  Otherwise, $ B$ wins.  If $ A$ begins, which player has a winning strategy?
1995 China Team Selection Test
Prove that the interval $\lbrack 0,1 \rbrack$ can be split into black and white intervals for any quadratic polynomial $P(x)$, such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals.  (Define the weight of the interval $\lbrack a,b \rbrack$ as $P(b) - P(a)$.)



Does the same result hold with a degree 3 or degree 5 polynomial?
1994 China Team Selection Test
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
1994 China Team Selection Test
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression.  If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.



a.) Find the smallest $s \in \mathbb{N}$  such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.



b.) Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
1994 China Team Selection Test
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
1994 China Team Selection Test
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$.  Prove that $\prod_{i=1}^{n}\frac {r_i s_i

t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
1994 China Team Selection Test
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.
1994 China Team Selection Test
For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$.



I. Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons.



II. Find the smallest possible value of $m$.
1993 China Team Selection Test
For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$
1993 China Team Selection Test
Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$
1993 China Team Selection Test
A graph $G=(V,E)$ is given. If at least $n$ colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''$n-$colored''. Prove that for any $n \in \mathbb{N}$, there is a $n-$colored graph without triangles.
1993 China Team Selection Test
Find all integer solutions to $2 x^4 + 1 = y^2.$
1993 China Team Selection Test
Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.
1993 China Team Selection Test
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
1992 China Team Selection Test
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.
1992 China Team Selection Test
Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds:



$$\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.$$
1992 China Team Selection Test
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
1992 China Team Selection Test
A triangle $ABC$ is given in the plane with $AB = \sqrt{7},$ $BC = \sqrt{13}$ and $CA = \sqrt{19},$ circles are drawn with centers at $A,B$ and $C$ and radii $\frac{1}{3},$ $\frac{2}{3}$ and $1,$ respectively. Prove that there are points $A',B',C'$ on these three circles respectively such that triangle $ABC$ is congruent to triangle $A'B'C'.$
1992 China Team Selection Test
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
1992 China Team Selection Test
For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have



$$\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},$$ where $S_k = \sum^k_{i=1} a_i.$
1991 China Team Selection Test
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have

$$f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.$$
1991 China Team Selection Test
For $i = 1,2, \ldots, 1991$, we choose $n_i$ points and write number $i$ on them (each point has only written one number on it). A set of chords are drawn such that:



(i) They are pairwise non-intersecting.

(ii) The endpoints of each chord have distinct numbers.



If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers $n_1, n_2, \ldots, n_{1991}$ must satisfy for this to be possible.
1991 China Team Selection Test
$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be good. Let the number of good circles be $n$; find all possible values of $n$.
1991 China Team Selection Test
We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$
1991 China Team Selection Test
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions:



(1) $f(0) = 0, f(1) = 1,$

(2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$



Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$
1991 China Team Selection Test
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called excentric. The excentricity of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.
1990 China Team Selection Test
In a wagon, every $m \geq 3$ people have exactly one common friend. (When $A$ is $B$'s friend, $B$ is also $A$'s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.
1990 China Team Selection Test
Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin $O$ that cuts them both, then these polygons are called "properly placed". Find the least $m \in \mathbb{N}$, such that for any group of properly placed polygons, $m$ lines can drawn through $O$ and every polygon is cut by at least one of these $m$ lines.
1990 China Team Selection Test
In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And



(i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$.



(ii) $a \circ b \neq b \circ a$ when $a \neq b$.



Prove that:



a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$.



b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.
1990 China Team Selection Test
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.
1990 China Team Selection Test
Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that:



$ \left(a + b\right) \cdot \left(\frac {1}{a} + \frac {1}{b} + \frac {1}{c} \right) \geq 4 + \frac {1}{\sin\left(\frac {C}{2}\right)}.$
1990 China Team Selection Test
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$
1990 China Team Selection Test
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
1990 China Team Selection Test
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
1989 China Team Selection Test
A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.
1989 China Team Selection Test
Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$
1989 China Team Selection Test
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
1989 China Team Selection Test
Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.
1989 China Team Selection Test
Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?
1989 China Team Selection Test
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$.



Alternative formulation. Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.
1989 China Team Selection Test
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
1989 China Team Selection Test
$\forall n \in \mathbb{N}$, $P(n)$ denotes the number of the partition of $n$ as the sum of positive integers (disregarding the order of the parts), e.g. since $4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4$, so $P(4)=5$. "Dispersion" of a partition denotes the number of different parts in that partitation. And denote $q(n)$ is the sum of all the dispersions, e.g. $q(4)=1+2+2+1+1=7$. $n \geq 1$. Prove that



(1) $q(n) = 1 + \sum^{n-1}_{i=1} P(i).$



(2) $1 + \sum^{n-1}_{i=1} P(i) \leq \sqrt{2} \cdot n \cdot P(n)$.
1988 China Team Selection Test
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:



$$A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.$$



Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
1988 China Team Selection Test
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying



(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.



(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
1988 China Team Selection Test
In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.
1988 China Team Selection Test
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.



(i) Find $r_5$.

(ii) Find $r_7$.

(iii) Find $r_k$ for $k \in \mathbb{N}$.
1988 China Team Selection Test
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
1988 China Team Selection Test
Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.
1988 China Team Selection Test
A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.
1988 China Team Selection Test
There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.
1987 China Team Selection Test
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:



i. each has $k$ elements;

ii. $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)

iii. the union of the $5$ sets has exactly $f(k)$ elements.



b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.
1987 China Team Selection Test
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
1987 China Team Selection Test
Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.
1987 China Team Selection Test
Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle $A$ inside it with maximum area (over all posible rectangles). Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure. Find the smallest lamda such that it works for all convex figures.
1987 China Team Selection Test
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
1987 China Team Selection Test
Let $ G$ be a simple graph with $ 2 \cdot n$ vertices and $ n^{2}+1$ edges. Show that this graph $ G$ contains a $ K_{4}-\text{one edge}$, that is, two triangles with a common edge.
1986 China Team Selection Test
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
1986 China Team Selection Test
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:



i) For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k = 1} a_k \cdot x_k \leq \sum^{n}_{k = 1} b_k \cdot x_k,$



ii) We have $ \sum^{s}_{k = 1} a_k \leq \sum^{s}_{k = 1} b_k$ for every $ s\in\left\{1,2,...,n-1\right\}$ and $ \sum^{n}_{k = 1} a_k = \sum^{n}_{k = 1} b_k$.
1986 China Team Selection Test
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:



I. There exists positive integer $k$ for which $A_{k+1}=A_k$.

II. Find such $A_k$ for $19^{86}.$
1986 China Team Selection Test
Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that:



$\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$).



Edited by orl.
1986 China Team Selection Test
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.
1986 China Team Selection Test
Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that:



i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}.

ii) The same as above replacing "area" for "perimeter".
1986 China Team Selection Test
Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that:



i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$



ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.
1986 China Team Selection Test
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.



i. Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.

ii. Prove that the $3 \cdot k - 1$ cannot be improved.
2016 Croatia Team Selection Test
Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that

$$  (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$
2016 Croatia Team Selection Test
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A move consists of choosing a row or a column and changing the color of every cell in the chosen row or column.

What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?
2016 Croatia Team Selection Test
Let $P$ be a point inside a triangle $ABC$ such that

$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
2016 Croatia Team Selection Test
Find all pairs $(p,q)$ of prime numbers such that

$$ p(p^2 - p - 1) = q(2q + 3) .$$
2016 Croatia Team Selection Test
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$:

$$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$
2016 Croatia Team Selection Test
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.

Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
2016 Croatia Team Selection Test
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.
2016 Croatia Team Selection Test
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.

Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
2011 Croatia Team Selection Test
Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality

$$\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.$$
2011 Croatia Team Selection Test
There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
2011 Croatia Team Selection Test
Triangle $ABC$ is given with its centroid $G$ and cicumcentre $O$ is such  that $GO$ is perpendicular to $AG$. Let $A'$ be the second intersection of $AG$  with circumcircle of triangle $ABC$. Let $D$ be the intersection of lines  $CA'$ and $AB$ and $E$ the intersection of lines $BA'$ and $AC$. Prove that the circumcentre of triangle $ADE$ is on the circumcircle of triangle $ABC$.
2011 Croatia Team Selection Test
We define the sequence $x_n$ so that

$$x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.$$

Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.
2011 Croatia Team Selection Test
We define a sequence $a_n$ so that $a_0=1$ and

$$a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases}  $$

for all postive integers $n$.

Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
2011 Croatia Team Selection Test
There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$