Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Italy TST
Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that

$$DE\le\frac{1}{8}(AB+BC+CA) $$
1999 Italy TST
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. $$

(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$  f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.$$
1999 Italy TST
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that



i) $|A_i|=3$ for each $i=1,\ldots ,m$.



ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.



Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
1998 Italy TST
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
1998 Italy TST
New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?
1998 Italy TST
Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.
1997 Italy TST
Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that $$x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.$$Find all possible values of $ t$ such that $xyz+t=0$.
1997 Italy TST
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
1997 Italy TST
Determine all triples $(x,y, p)$ with $x$, $y$ positive integers and $p$ a prime number verifying the equation $p^x -y^p = 1$.
1997 Italy TST
There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?
1996 Italy TST
1-Let $A$ and $B$ be two diametrically opposite points on a circle with radius $1$. Points $P_1,P_2,...,P_n$ are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of $P_1,P_2,...,P_n$ from $A$ and $B$, respectively. Show that at least one of the numbers $a$ and $b$ does not exceed $\sqrt{2}$
1996 Italy TST
2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$  ,:......., $A_n\setminus \{x\}$ are all distinct.
1996 Italy TST
3.Let ABCD be a parallelogram with side AB longer than AD and acute angle

$\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is

the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.
1996 Italy TST
4.4. Prove that there exists a set X of 1996 positive integers with the following properties:

(i) the elements of X are pairwise coprime;

(ii) all elements of X and all sums of two or more distinct elements of X are

composite numbers
1995 Italy TST
Determine all triples $(x,y,z)$ of integers greater than $1$ with the property that $x$ divides $yz-1$, $y$ divides $zx-1$ and $z$ divides $xy-1$.
1995 Italy TST
Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?
1995 Italy TST
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions

$$\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}$$

Prove that $f(x+1)=f(x)+1$ for all real $x$.
1995 Italy TST
In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.
1994 Italy TST
Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.
1994 Italy TST
Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.
1994 Italy TST
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
1994 Italy TST
Let $X$ be a set of $n$ elements and $k$ be a positive integer.

Consider the family $S_k$ of all $k$-tuples $(E_1,...,E_k)$ with $E_i \subseteq X$ for each $i$.

Evaluate the sums $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cap ... \cap E_k|$ and  $\sum_{(E_1,...,E_k) \in S_k }|E_1 \cup ... \cup E_k|$
1993 Italy TST
Let $x_1,x_2,...,x_n$ ($n \ge 2$) be positive numbers with the sum $1$. Prove that

$$\sum_{i=1}^{n} \frac{1}{\sqrt{1-x_i}} \ge  n\sqrt{\frac{n}{n-1}} $$
1993 Italy TST
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.

Show that $p = q$.
1993 Italy TST
Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$
1993 Italy TST
An $m \times n$ chessboard with $m,n \ge 2$ is given.

Some dominoes are placed on the chessboard so that the following conditions are satisfied:

(i) Each domino occupies two adjacent squares of the chessboard,

(ii) It is not possible to put another domino onto the chessboard without overlapping,

(iii) It is not possible to slide a domino horizontally or vertically without overlapping.

Prove that the number of squares that are not covered by a domino is less than $\frac15 mn$.
2020 Kosovo Team Selection Test
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$ satisfy,

$$f\left(x+yf(x+y)\right)=y^2+f(x)f(y)$$

Proposed by Dorlir Ahmeti, Kosovo
2020 Kosovo Team Selection Test
Let $p$ be an odd prime number. Ana and Ben are playing a game with alternate moves as follows: in each move, the player which has the turn choose a number, which was not choosen before by any of the player, from the set $\{1,2,...,2p-3,2p-2\}$. This process continues until no number is left. After the end of the process, each player create the number by taking the product of the choosen numbers and then add 1. We say a player wins if the number that did create is divisible by $p$, while the number that did create the opponent it is not divisible by $p$, otherwise we say the game end in a draw. Ana start first move.

Does it exist a strategy for any of the player to win the game?



Proposed by Dorlir Ahmeti, Kosovo
2020 Kosovo Team Selection Test
Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$



Proposed by Viktor Ahmeti and Leart Ajvazaj, Kosovo
2020 Kosovo Team Selection Test
Prove that for all positive integers $m$ and $n$ the following inequality hold:

$$\pi(m)-\pi(n)\leq\frac{(m-1)\varphi(n)}{n}$$When does equality hold?



Proposed by Shend Zhjeqi and Dorlir Ahmeti, Kosovo
2019 Kosovo Team Selection Test
There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.
2019 Kosovo Team Selection Test
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$

$$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$
2019 Kosovo Team Selection Test
Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.
2019 Kosovo Team Selection Test
Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is

$$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.
2019 Kosovo Team Selection Test
$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0}  \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$
2017 Kosovo Team Selection Test
Find all positive integers $(a, b)$, such that  $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.
2017 Kosovo Team Selection Test
Prove that there doesn't exist any function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that :

$f(f(n-1)=f(n+1)-f(n)$, for every natural $n\geq2$
2017 Kosovo Team Selection Test
If $a$ and $b$ are positive real numbers with sum $3$, and $x, y, z$ positive real numbers with product $1$, prove that :

$(ax+b)(ay+b)(az+b)\geq 27$
2017 Kosovo Team Selection Test
For every $n \in \mathbb{N}_{0}$, prove that

$\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}$
2017 Kosovo Team Selection Test
Given triangle $ABC$. Let $P$, $Q$, $R$, be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$, $BC$, $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$. Prove that $AP'$, $BQ'$, and $CR'$ are concurrent.
2016 Kosovo Team Selection Test
Solve equation in real numbers



$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$
2016 Kosovo Team Selection Test
Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .
2016 Kosovo Team Selection Test
If quadratic equations $x^2+ax+b=0$ and $x^2+px+q=0$ share one similar root then find quadratic equation for which has roots of other roots of both quadratic equations .
2016 Kosovo Team Selection Test
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied



$f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$



If $g(x)=f(x)-x+1$ then find $g(2016)$ .
2016 Kosovo Team Selection Test
Let be $ABC$ an acute triangle with $|AB|>|AC|$ . Let be $D$ point in side $AB$ such that $\angle ACD=\angle CBD$ . Let be $E$ the midpoint of segment $BD$ and $S$ let be the circumcenter of triangle $BCD$ . Show that points $A,E,S$ and $C$ lie on a circle .
2015 Kosovo Team Selection Test
a)Prove that for every n,natural number exist natural numbers a and b such that

$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$

b)Using first equation prove that for every n exist m such that

$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
2015 Kosovo Team Selection Test
Prove that circle l(0,2) with equation $x^2+y^2=4$ contains infinite points with rational coordinates
2015 Kosovo Team Selection Test
It's given system of equations

$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$

$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$

..........

$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$

such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
2015 Kosovo Team Selection Test
Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$. If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$, with vertices from the set $S$.
2015 Kosovo Team Selection Test
In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular.

a)Prove that midpoints M,N,P,Q of AD,AB,BC,CD form a rectangular

b)If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO
2012 Kosovo Team Selection Test
A student had $18$ papers. He seleced some of these papers, then he cut each of them in $18$ pieces.He took these pieces and selected some of them, which he again cut in $18$ pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got $2012$ pieces .Prove that the student was wrong during the counting.
2012 Kosovo Team Selection Test
Find all three digit numbers, for which the sum of squares of each digit is $90$ .
2012 Kosovo Team Selection Test
If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that

$$4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)$$
2012 Kosovo Team Selection Test
Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs
2012 Kosovo Team Selection Test
Prove that the equation

$$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$

has infinitly many natural solutions
2011 Kosovo Team Selection Test
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds:



$${

\sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot  \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2}

}$$
2011 Kosovo Team Selection Test
Prove that the lines joining the middle-points of non-adjacent sides of an convex quadrilateral and the line joining the middle-points of diagonals, are concurrent. Prove that the intersection point is the middle point of the three given segments.
2011 Kosovo Team Selection Test
Let $n$ be a natural number, for which we define $S(n)=\{1+g+g^2+...+g^{n-1}|g\in{\mathbb{N}},g\geq2\}$



$a)$ Prove that: $S(3)\cap S(4)=\varnothing$



$b)$ Determine: $S(3)\cap S(5)$
2011 Kosovo Team Selection Test
From the number $7^{1996}$ we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.
2011 Kosovo Team Selection Test
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $ \forall x\notin\{-1,1\}$ holds:



$$\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}$$
2006 Lithuania Team Selection Test
Let $a_1, a_2, \dots, a_n$ be positive real numbers, whose sum is $1$. Prove that $$ \frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\dots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+\frac{a_n^2}{a_n+a_1}\ge \frac{1}{2} $$
2006 Lithuania Team Selection Test
Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.
2006 Lithuania Team Selection Test
Inside a convex quadrilateral $ABCD$ there is a point $P$ such that the triangles $PAB, PBC, PCD, PDA$ have equal areas. Prove that the area of $ABCD$ is bisected by one of the diagonals.
2006 Lithuania Team Selection Test
Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.
2006 Lithuania Team Selection Test
Does the bellow depicted figure fit into a square $5\times5$.
2021 Moldova Team Selection Test
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.



Brazitikos Silouanos, Greece
2021 Moldova Team Selection Test
Prove that if $p$ and $q$ are two prime numbers, such that

$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p$,

then $p=q$.
2021 Moldova Team Selection Test
Acute triangle $ABC$ with $AB>BC$ is inscribed in circle $\Omega$. Points $D$ and $E$, that lie on $(BC)$ and $(AB)$ are the feet of altitudes from $A$ and $C$ in triangle $ABC$, and $M$ is the midpoint of the segment $DE$. Half-line $(AM$ intersects the circle $\Omega$ for the second time in $N$. Show that the circumcenter of triangle $MDN$ lies on the line $BC$.
2021 Moldova Team Selection Test
Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.
2021 Moldova Team Selection Test
Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.
2021 Moldova Team Selection Test
There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last eight players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)
2021 Moldova Team Selection Test
Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Show that

$\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}$.

When does the equality take place?
2021 Moldova Team Selection Test
Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$.

$Turkey$
2021 Moldova Team Selection Test
Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of

$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}$.
2021 Moldova Team Selection Test
On a board there are written the integers from $1$ to $119$. Two players, $A$ and $B$, make a move by turn. A $move$ consists in erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player $A$ makes the first move. Find the highest integer $k$, such that the player $A$ can be sure that his score is not smaller than $k$.
2021 Moldova Team Selection Test
In a convex quadrilateral $ABCD$ the angles $BAD$ and $BCD$ are equal. Points $M$ and $N$ lie on the sides $(AB)$ and $(BC)$ such that the lines $MN$ and $AD$ are parallel and $MN=2AD$. The point $H$ is the orthocenter of the triangle $ABC$ and the point $K$ is the midpoint of $MN$. Prove that the lines $KH$ and $CD$ are perpendicular.
2021 Moldova Team Selection Test
Prove that $n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$ for every integer $n \geq 3$.
2020 Moldova Team Selection Test
All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2020 Moldova Team Selection Test
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$
2020 Moldova Team Selection Test
Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?
2020 Moldova Team Selection Test
Let $\Delta ABC$ be an acute triangle and $H$ its orthocenter. $B_1$ and $C_1$ are the feet of heights from $B$ and $C$, $M$ is the midpoint of $AH$. Point $K$ is on the segment $B_1C_1$, but isn't on line $AH$. Line $AK$ intersects the lines $MB_1$ and $MC_1$ in $E$ and $F$, the lines $BE$ and $CF$ intersect at $N$. Prove that $K$ is the orthocenter of $\Delta NBC$.
2020 Moldova Team Selection Test
Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$On interval $\left[0,\frac{\pi}{4}\right).$
2020 Moldova Team Selection Test
Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$
2020 Moldova Team Selection Test
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place

$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1$.
2020 Moldova Team Selection Test
In $\Delta ABC$ the angles $ABC$ and $ACB$ are acute. Let $M$ be the midpoint of $AB$. Point $D$ is on the half-line $(CB$ such that $ B \in (CD)$ and $\angle DAB= \angle BCM$. Perpendicular from $B$ to line $CD$ intersects the line bisector of $AB$ in $E$. Prove that $DE$ and $AC$ are perpendicular.
2020 Moldova Team Selection Test
Let $\Delta ABC$ be an acute triangle and $\Omega$ its circumscribed circle, with diameter $AP$. Points $E$ and $F$ are the orthogonal projections from $B$ on $AC$ and $AP$, points $M$ and $N$ are the midpoints of segments $EF$ and $CP$. Prove that $\angle BMN=90$.
2020 Moldova Team Selection Test
Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$
2020 Moldova Team Selection Test
Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy

$$f(\sin{x})+f(\cos{x})=2020$$for any real number $x.$
2020 Moldova Team Selection Test
In a chess tournament each player played one match with every other player. It is known that all players have different scores. The player who is on the last place got $k$ points. What is the smallest number of wins that the first placed player got? (For the win $1$ point is given, for loss $0$ and for a draw both players get $0,5$ points.)
2019 Moldova Team Selection Test
Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$.

(A natural number is a positive integer)
2019 Moldova Team Selection Test
Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$.

(A natural number is a positive integer)
2019 Moldova Team Selection Test
On the table there are written numbers $673, 674, \cdots, 2018, 2019.$ Nibab chooses arbitrarily three numbers $a,b$ and $c$, erases them and writes the number $\frac{\min(a,b,c)}{3}$, then he continues in an analogous way. After Nibab performed this operation $673$ times, on the table remained a single number $k$. Prove that $k\in(0,1).$