Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Moldova Team Selection Test
Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.
2019 Moldova Team Selection Test
Point $H$ is the orthocenter of the scalene triangle $ABC.$ A line, which passes through point $H$, intersect the sides $AB$ and $AC$ at points $D$ and $E$, respectively, such that $AD=AE.$ Let $M$ be the midpoint of side $BC.$ Line $MH$ intersects the circumscribed circle of triangle $ABC$ at point $K$, which is on the smaller arc $AB$. Prove that Nibab can draw a circle through $A, D, E$ and $K.$
2019 Moldova Team Selection Test
Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$.

Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$
2019 Moldova Team Selection Test
Let $P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0$ be a polynomial with all positive coefficients. Prove that there exists a permutation $(b_{2n+1},b_{2n},...,b_1,b_0)$ of numbers $(a_{2n+1},a_{2n},...,a_1,a_0)$ such that the polynomial $Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0$ has exactly one real root.
2019 Moldova Team Selection Test
For any positive integer $k$ denote by $S(k)$ the number of solutions $(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+$ of the system

$$\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}$$where $d$ is the greatest common divisor of positive integers $x$ and $y.$ Determine $S(k)$ as a function of $k$. (Here $\lceil z\rceil$ denotes the smalles integer number which is bigger or equal than $z.$)
2019 Moldova Team Selection Test
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
2019 Moldova Team Selection Test
Let $n\ge 2,$ be a positive integer. Numbers $\{1,2,3, ...,n\}$ are written in a row in an arbitrary order. Determine the smalles positive integer $k$ with the property: everytime it is possible to delete $k$ numbers from those written on the table, such that the remained numbers are either in an increasing or decreasing order.
2019 Moldova Team Selection Test
Let $p\ge 5$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n\le \frac{p+1}{2}$ for which $p$ divides $2^n\cdot 3^m-1.$
2018 Moldova Team Selection Test
Let $x,y,z \in\mathbb{Q}$,such that $(x+y+z)^3=9(x^2y+y^2z+z^2x).$ Prove that $x=y=z$
2018 Moldova Team Selection Test
The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$

Prove that

$$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$for any $n\in\mathbb{N}$
2018 Moldova Team Selection Test
A pupil is writing on a board positive integers $x_0,x_1,x_2,x_3...$ after the following algorithm which implies arithmetic progression $3,5,7,9...$.Each term of rank $k\ge2$ is a difference between the product of the last number on the board and the term of arithmetic progression of rank $k$ and the last but one term on the bord with the sum of the terms of the arithemtic progression with ranks less than $k$.If $x_0=0 $ and $x_1=1$ find $x_n$ according to n.
2018 Moldova Team Selection Test
Let $n,  \in \mathbb {N^*} , n\ge 3$

a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has  a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $



b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .
2018 Moldova Team Selection Test
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that

$$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$
2018 Moldova Team Selection Test
Let the triangle $ABC $ with $m (\angle ABC)=60^{\circ} $ and $m (\angle BAC)=40^{\circ}$ . Points $D $ and $E $ are on the sides $(AB) $ and $(AC) $ such that $m (\angle DCB )=70^{\circ}$ and $m (\angle EBC)=40^{\circ}$ . $BE$ and $CD$ intersect in  $F $ . Prove that $BC $ and $AF $ are perpendicular.
2018 Moldova Team Selection Test
Let the set  $A=${$ 1,2,3, \dots ,48n+24$ } , where $  n \in \mathbb {N^*}$ . Prove that there exist a subset $B $ of $A $ with $24n+12$ elements with the property : the sum of the squares of the elements of the set $B $ is equal to the sum of the squares of the elements of the set $A$ \ $B $ .
2018 Moldova Team Selection Test
The positive integers $a $ and  $b $ satisfy the sistem $\begin {cases} a_{10} +b_{10} = a \\a_{11}+b_{11 }=b \end {cases} $ where  $ a_1 <a_2 <\dots $ and  $ b_1 <b_2 <\dots $ are the positive divisors of $a $ and $b$ .

Find $a$ and $b $ .
2018 Moldova Team Selection Test
The positive real numbers $a,b, c,d$ satisfy the equality $ \frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} +  \frac{ 1}{d+1} = 3 $ . Prove the inequality $\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3} $.
2018 Moldova Team Selection Test
Let $\Omega $ be the circumcincle of the quadrilateral $ABCD $ , and $E $ the intersection point of the diagonals  $AC $ and $BD $ . A line passing through  $E $ intersects  $AB $ and $BC$ in points $P $ and $Q $ . A circle ,that is passing through point $D $ , is tangent to the line $PQ $ in point $E $ and intersects $\Omega$ in point $R $ , different from $D $ . Prove that the points  $B,P,Q,$ and $R $ are concyclic .
2018 Moldova Team Selection Test
Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that $${p \choose 1}+{p \choose 2}+\cdots+{p \choose k}$$ is divisible by $p^{2}$.
2017 Moldova Team Selection Test
Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as:

$$a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},$$where $A_{m}^{k}$ are defined by $$A_{m}^{k}=\binom{m}{k}\cdot k!.$$Prove that $$a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.$$
2017 Moldova Team Selection Test
Let $$f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$$be a polynomial with real coefficients which satisfies $$a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.$$Prove that for every complex root $z$ of this polynomial, we have $|z|\leq 1$.
2017 Moldova Team Selection Test
Let $\omega$ be the circumcircle of the acute nonisosceles triangle $\Delta ABC$. Point $P$ lies on the altitude from $A$. Let $E$ and $F$ be the feet of the altitudes from P to $CA$, $BA$ respectively. Circumcircle of triangle $\Delta AEF$ intersects the circle $\omega$ in $G$, different from $A$. Prove that the lines $GP$, $BE$ and $CF$ are concurrent.
2017 Moldova Team Selection Test
Determine all natural numbers $n$ of the form $n=[a,b]+[b,c]+[c,a]$ where $a,b,c$ are positive integers and $[u,v]$ is the least common multiple of the integers $u$ and $v$.
2017 Moldova Team Selection Test
Find all continuous functions  $f : R \rightarrow R$ such, that $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$ for any real numbers $x$ and $y$
2017 Moldova Team Selection Test
Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that

$$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$
2017 Moldova Team Selection Test
Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality:

$$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.
2017 Moldova Team Selection Test
At a summer school there are $7$ courses. Each participant was a student in at least one course, and each course was taken by exactly $40$ students. It is known that for each $2$ courses there were at most $9$ students who took them both. Prove that at least $120$ students participated at this summer school.
2017 Moldova Team Selection Test
Let $$P(X)=a_{0}X^{n}+a_{1}X^{n-1}+\cdots+a_{n}$$be a polynomial with real coefficients such that $a_{0}>0$ and $$a_{n}\geq a_{i}\geq 0,$$for all $i=0,1,2,\ldots,n-1.$ Prove that if $$P^{2}(X)=b_{0}X^{2n}+b_{1}X^{2n-1}+\cdots+b_{n-1}X^{n+1}+\cdots+b_{2n},$$then $P^2(1)\geq 2b_{n-1}.$
2017 Moldova Team Selection Test
Let $p$ be an odd prime. Prove that the number

$$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$is divisible by $20p$.
2017 Moldova Team Selection Test
Find all ordered pairs of nonnegative integers $(x,y)$ such that

$$x^4-x^2y^2+y^4+2x^3y-2xy^3=1.$$
2017 Moldova Team Selection Test
There are $75$ points in the plane, no three collinear. Prove that the number of acute triangles is no more than $70\%$ from the total number of triangles with vertices in these points.
2016 Moldova Team Selection Test
If $x_1,x_2,...,x_n>0 $ and $x_1^2+x_2^2+...+x_n^2=\dfrac{1}{n}$,prove that $\sum x_i+\sum \dfrac{1}{x_i \cdot x_{i+1}} \ge n^3+1.$
2016 Moldova Team Selection Test
Let $p$ be a prime number of the form $4k+1$. Show that $$\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.$$
2016 Moldova Team Selection Test
The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows:

$P_{1}(X)=2X$

$P_{2}(X)=2(X^2+1)$

$P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$.

Find all $n$ for which $X^2+1\mid P_{n}(X)$
2016 Moldova Team Selection Test
Let $n\in \mathbb{Z}_{> 0}$. The  set $S$ contains all positive integers written in decimal form that simultaneously satisfy the following conditions:



each element of $S$ has exactly $n$ digits;

each element of $S$ is divisible by $3$;

each element of $S$ has all its digits from the set $\{3,5,7,9\}$



Find $\mid S\mid$
2016 Moldova Team Selection Test
Let $\Omega$ and $O$ be the circumcircle of acute triangle $ABC$ and its center, respectively. $M\ne O$ is an arbitrary point in the interior of $ABC$ such that $AM$, $BM$, and $CM$ intersect $\Omega$ at $A_{1}$, $B_{1}$, and $C_{1}$, respectiuvely. Let $A_{2}$, $B_{2}$, and $C_{2}$ be the circumcenters of $MBC$, $MCA$, and $MAB$, respectively. It is to be proven that $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C{2}$ concur.
2016 Moldova Team Selection Test
Let us have $n$ ( $n>3$) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number $d$ such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by $d$.
2016 Moldova Team Selection Test
Let $\alpha \in \left( 0, \dfrac{\pi}{2}\right)$.Find the minimum value of the expression

$$ P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .$$
2016 Moldova Team Selection Test
Let $A_{1}A_{2} \cdots A_{14}$ be a regular $14-$gon. Prove that $A_{1}A_{3}\cap A_{5}A_{11}\cap A_{6}A_{9}\ne \emptyset$.
2016 Moldova Team Selection Test
Let $ABCD$ be a cyclic quadrilateral. Circle with diameter $AB$ intersects $CA$, $CB$, $DA$, and $DB$ in $E$, $F$, $G$, and $H$, respectively (all different from $A$ and $B$). The lines $EF$ and $GH$ intersect in $I$. Prove that the bisector of $\angle GIF$ and the line $CD$ are perpendicular.
2015 Moldova Team Selection Test
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.
2015 Moldova Team Selection Test
Consider a triangle $\triangle ABC$, let the incircle centered at $I$ touch the sides $BC,CA,AB$ at points $D,E,F$ respectively. Let the angle bisector of $\angle BIC$ meet $BC$ at $M$, and the angle bisector of $\angle EDF$ meet $EF$ at $N$. Prove that $A,M,N$ are collinear.
2015 Moldova Team Selection Test
Let $p$ be a fixed odd prime. Find the minimum positive value of $E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}$ where $x,y \in \mathbb{Z}_{+}$.
2015 Moldova Team Selection Test
In how many ways can we color exactly $k$ vertices of an $n$-gon in red such that any $2$ consecutive vertices are not both red. (Vertices are considered to be labeled)
2015 Moldova Team Selection Test
Let  $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$.

Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.
2015 Moldova Team Selection Test
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove the following inequality:

$a^3+b^3+c^3+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2} \geq \frac{9}{2}$.
2015 Moldova Team Selection Test
Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.
2015 Moldova Team Selection Test
Consider a positive integer $n$ and $A = \{ 1,2,...,n \}$. Call a subset $X \subseteq A$ perfect if $|X| \in X$. Call a perfect subset $X$ minimal if it doesn't contain another perfect subset. Find the number of minimal subsets of $A$.
2015 Moldova Team Selection Test
Find all polynomials $P(x)$ with real coefficients which satisfies

$P(2015)=2025$ and $P(x)-10=\sqrt{P(x^{2}+3)-13}$ for every $x\ge 0$ .
2015 Moldova Team Selection Test
Prove the equality:

$\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}$ .
2015 Moldova Team Selection Test
The tangents to the inscribed circle of $\triangle ABC$, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points $M,N,P,Q,R,S$ such that $M,S\in (AB)$, $N,P\in (AC)$, $Q,R\in (BC)$. The interior angle bisectors of $\triangle AMN$, $\triangle BSR$ and $\triangle CPQ$, from points $A,B$ and respectively $C$ have lengths $l_{1}$ , $l_{2}$  and $l_{3}$ .

Prove the inequality: $\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}$ where $p$ is the semiperimeter of $\triangle ABC$ .
2015 Moldova Team Selection Test
Let $n$ and $k$ be positive integers, and let be the sets $X=\{1,2,3,...,n\}$ and $Y=\{1,2,3,...,k\}$.

Let $P$ be the set of all the subsets of the set $X$. Find the number of functions $ f: P \to Y$ that satisfy $f(A \cap B)=\min(f(A),f(B))$ for all $A,B \in P$.
2014 Moldova Team Selection Test
Find all pairs of non-negative integers $(x,y)$ such that

$$\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.$$
2014 Moldova Team Selection Test
Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression:



$$E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.$$
2014 Moldova Team Selection Test
Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
2014 Moldova Team Selection Test
Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:

$$p(1)+p(2)+p(3)+...+p(999).$$
2014 Moldova Team Selection Test
Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.
2014 Moldova Team Selection Test
Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of

$E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$ .
2014 Moldova Team Selection Test
Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.
2014 Moldova Team Selection Test
Consider $n \geq 2$ distinct points in the plane $A_1,A_2,...,A_n$ . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?
2014 Moldova Team Selection Test
Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.
2014 Moldova Team Selection Test
Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$:

$f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,
2014 Moldova Team Selection Test
Let $\triangle ABC$ be a triangle with $\angle A$-acute. Let $P$ be a point inside $\triangle ABC$ such that $\angle BAP = \angle ACP$ and $\angle CAP =\angle ABP$. Let $M, N$ be the centers of the incircle of $\triangle ABP$ and $\triangle ACP$, and $R$ the radius of the circumscribed circle of $\triangle AMN$. Prove that $\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}. $
2014 Moldova Team Selection Test
On a circle $n \geq 1$ real numbers are written, their sum is $n-1$. Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise,  such that for any $k$ ($1\leq k \leq n$) $ x_1 + x_2+...+x_k \geq k-1$.
2013 Moldova Team Selection Test
For any positive real numbers $x,y,z$, prove that

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$
2013 Moldova Team Selection Test
We call a triangle $\triangle ABC$, $Q$-angled if $\tan\angle A,\tan\angle B,\tan\angle C \in \mathbb{Q}$, where $\angle A,\angle B ,\angle C$ are the interior angles of the triangle $\triangle ABC$.

$a)$ Prove that $Q$-angled triangles exist;

$b)$ Let triangle $\triangle ABC$ be $Q$-angled. Prove that for any non-negative integer $n$, numbers of the form

$E_n=\sin^n\angle A \sin^n\angle B \sin^n\angle C + \cos^n\angle A\cos^n\angle B\cos^n\angle C$ are rational.
2013 Moldova Team Selection Test
Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:

$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.

$a)$ Prove that the sequence is converging.

$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.
2013 Moldova Team Selection Test
Let $m$ be the number of ordered solutions $(a,b,c,d,e)$ satisfying:

$1)$ $a,b,c,d,e\in \mathbb{Z}^{+}$;

$2)$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$;

Prove that $m$ is odd.
2013 Moldova Team Selection Test
Let $a_n=1+n!(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!})$ for any $n\in \mathbb{Z}^{+}$. Consider $a_n$ points in the plane,no $3$ of them collinear.The segments between any $2$ of them are colored in one of $n$ colors. Prove that among them there exist $3$ points forming a monochromatic triangle.
2013 Moldova Team Selection Test
The diagonals of a trapezoid $ABCD$ with $AD \parallel BC$ intersect at point $P$. Point $Q$ lies between the parallel lines $AD$ and $BC$ such that the line $CD$ separates points $P$ and $Q$, and $\angle AQD=\angle CQB$. Prove that $\angle BQP = \angle DAQ$.
2013 Moldova Team Selection Test
Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$,

$(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$
2013 Moldova Team Selection Test
Let $A=20132013...2013$ be formed by joining $2013$, $165$ times. Prove that $2013^2 \mid A$.
2013 Moldova Team Selection Test
Find all pairs of real numbers $(x,y)$ satisfying

$\left\{\begin{array}{rl}

2x^2+xy &=1 \\ 

\frac{9x^2}{2(1-x)^4}&=1+\frac{3xy}{2(1-x)^2}

\end{array}\right.$
2013 Moldova Team Selection Test
Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.
2013 Moldova Team Selection Test
Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that $$ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. $$
2013 Moldova Team Selection Test
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?
2013 Moldova Team Selection Test
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.
2011 Moldova Team Selection Test
Find all real numbers $x, y$ such that:

$y+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}$
2011 Moldova Team Selection Test
Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality

$\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$
2011 Moldova Team Selection Test
Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.
2011 Moldova Team Selection Test
Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression:

$E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$
2011 Moldova Team Selection Test
Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.
2011 Moldova Team Selection Test
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:

$x+y+4=\frac{12x+11y}{x^2+y^2}$



$y-x+3=\frac{11x-12y}{x^2+y^2}$
2011 Moldova Team Selection Test
Let $ABC$ be a triangle with $\angle BAC=60$. Let $B_1$ and $C_1$ be the feet of the bisectors from $B$ and $C$. Let $A_1$ be the symmetrical of $A$ according to line $B_1C_1$. Prove that $A_1, B, C$ are colinear.
2011 Moldova Team Selection Test
Initially, on the blackboard are written all natural numbers from $1$ to $20$. A move consists of selecting $2$ numbers $a<b$ written on the blackboard such that their difference is at least $2$, erasing these numbers and writting $a+1$ and $b-1$ instead. What is the maximum numbers of moves one can perform?
2010 Moldova Team Selection Test
Find all $ 3$-digit numbers such that placing to the right side of the number its successor we get a $ 6$-digit number which is a perfect square.
2010 Moldova Team Selection Test
Prove that for any real number $ x$ the following inequality is true:

$ \max\{|\sin x|, |\sin(x+2010)|\}>\dfrac1{\sqrt{17}}$
2010 Moldova Team Selection Test
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC=3\angle CAD$, $ AB=CD$, $ \angle ACD=\angle CBD$. Find angle $ \angle ACD$
2010 Moldova Team Selection Test
Let $ n\geq6$ be a even natural number. Prove that any cube can be divided in $ \dfrac{3n(n-2)}4+2$ cubes.
2010 Moldova Team Selection Test
Let $ p\in\mathbb{R}_+$ and $ k\in\mathbb{R}_+$. The polynomial $ F(x)=x^4+a_3x^3+a_2x^2+a_1x+k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p+k)^4$
2010 Moldova Team Selection Test
Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of:

$ E=x_1+\dfrac{x_2}{\sqrt{1-x_1^2}}+\dfrac{x_3}{\sqrt{1-(x_1+x_2)^2}}+\cdots+\dfrac{x_n}{\sqrt{1-(x_1+x_2+\cdots+x_{n-1})^2}}$