Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Moldova Team Selection Test
Let $ ABC$ be an acute triangle. $ H$ is the orthocenter and $ M$ is the middle of the side $ BC$. A line passing through $ H$ and perpendicular to $ HM$ intersect the segment $ AB$ and $ AC$ in $ P$ and $ Q$. Prove that $ MP = MQ$
2009 Moldova Team Selection Test
Let $ m,n\in \mathbb{N}^*$. Find the least $ n$ for which exists $ m$, such that rectangle $ (3m + 2)\times(4m + 3)$ can be covered with $ \dfrac{n(n + 1)}{2}$ squares, among which exist $ n$ squares of length $ 1$, $ n - 1$ of length $ 2$, $ ...$, $ 1$ square of length $ n$. For the found value of $ n$ give the example of covering.
2009 Moldova Team Selection Test
Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i = \overline{1,n}$, such that $ \sum a_i = 1$. Prove that



$ \small{\dfrac{a_1^{2 - m} + a_2 + ... + a_{n - 1}}{1 - a_1} + \dfrac{a_2^{2 - m} + a_3 + ... + a_n}{1 - a_1} + ... + \dfrac{a_n^{2 - m} + a_1 + ... + a_{n - 2}}{1 - a_1}\ge n + \dfrac{n^m - n}{n - 1}}$
2009 Moldova Team Selection Test
Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}=CA_1 \cap BD$ and $ \{Q\}=DB_1\cap AC$. Prove that $ AC\perp PQ$.
2009 Moldova Team Selection Test
Let $ p$ be a prime divisor of $ n\ge 2$. Prove that there exists a set of natural numbers $ A = \{a_1,a_2,...,a_n\}$ such that product of any two numbers from $ A$ is divisible by the sum of any $ p$ numbers from $ A$.
2009 Moldova Team Selection Test
For any $ m \in \mathbb{N}^*$ solve the ecuation

$$ \left\{\left( x + \frac {1}{m}\right) ^3\right\} = x^3

$$
2009 Moldova Team Selection Test
Determine all functions $ f : [0; + \infty) \rightarrow [0; + \infty)$, such that

$$ f(x + y - z) + f(2\sqrt {xz}) + f(2\sqrt {yz}) = f(x + y + z)$$

for all $ x,y,z \in [0; + \infty)$, for which $ x + y\ge z$.
2009 Moldova Team Selection Test
A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.
2009 Moldova Team Selection Test
Let $ m$ and $ n$ be two nonzero natural numbers. In every cell $ 1 \times 1$ of the rectangular table $ 2m \times 2n$ are put signs $ +$ or $ -$. We call cross an union of all cells which are situated in a line and in a column of the table. Cell, which is situated at the intersection of these line and column is called center of the cross. A transformation is defined in the following way: firstly we mark all points with the sign $ -$. Then consecutively, for every marked cell we change the signs in the cross, whose center is the choosen cell. We call a table accesible if it can be obtained from another table after one transformation.

Find the number of all accesible tables.
2009 Moldova Team Selection Test
Points $ X$, $ Y$ and $ Z$ are situated on the sides $ (BC)$, $ (CA)$ and $ (AB)$ of the triangles $ ABC$, such that triangles $ XYZ$ and $ ABC$ are similiar. Prove that circumcircle of $ AYZ$ passes through a fixed point.
2009 Moldova Team Selection Test
Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$.



a) Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios.



b) Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.
2009 Moldova Team Selection Test
Weightlifter Ruslan has just finished the exercise with a weight, which has $ n$ small weights on one side and $ n$ on the another. At each stage he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $ k$. What is the minimal number of stages (in function if $ n$ and $ k$), which Ruslan need to take off all weights..
2009 Moldova Team Selection Test
let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality:



$ \dfrac{60a^2-1}{4xy+5z}+\dfrac{60b^2-1}{4yz+5x}+\dfrac{60c^2-1}{4zx+5y}\geq 12$
2009 Moldova Team Selection Test
Let $ ABCD$ be a trapezoid with $ AB\parallel CD$. Exterior equilateral triangles $ ABE$ and $ CDF$ are constructed. Prove that lines $ AC$, $ BD$ and $ EF$ are concurrent.
2009 Moldova Team Selection Test
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)+2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.
2009 Moldova Team Selection Test
The sequence $ (a_n)_{n \in \mathbb{N}}$ is defined as follows:

$$ a_n = \dfrac{2}{3 + 1} + \dfrac{2^2}{3^2 + 1} + \dfrac{2^3}{3^4 + 1} + \ldots + \dfrac{2^{n + 1}}{3^{2^n} + 1}

$$

Prove that $ a_n < 1$ for any $ n \in \mathbb{N}$
2009 Moldova Team Selection Test
Let $ X$ be a group of people, where any two people are friends or enemies. Each pair of friends from $ X$ doesn't have any common friends, and any two enemies have exactly two common friends. Prove that each person from $ X$ has the same number of friends as others.
2008 Moldova Team Selection Test
Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3+y^3-3xy=p-1$.
2008 Moldova Team Selection Test
We say the set $ \{1,2,\ldots,3k\}$ has property $ D$ if it can be partitioned into disjoint triples so that in each of them a number equals the sum of the other two.



(a) Prove that $ \{1,2,\ldots,3324\}$ has property $ D$.



(b) Prove that $ \{1,2,\ldots,3309\}$ hasn't property $ D$.
2008 Moldova Team Selection Test
Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.
2008 Moldova Team Selection Test
A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)=\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.
2008 Moldova Team Selection Test
Find all solutions $ (x,y)\in \mathbb{R}\times\mathbb R$ of the following system: $ \begin{cases}x^3 + 3xy^2 = 49, \\
x^2 + 8xy + y^2 = 8y + 17x.\end{cases}$
2008 Moldova Team Selection Test
Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1+a_2+\ldots+a_n\le\frac n2$. Find the minimal value of

$ \sqrt{a_1^2+\frac1{a_2^2}}+\sqrt{a_2^2+\frac1{a_3^2}}+\ldots+\sqrt{a_n^2+\frac1{a_1^2}}$.
2008 Moldova Team Selection Test
Let $ \omega$ be the circumcircle of $ ABC$ and let $ D$ be a fixed point on $ BC$, $ D\neq B$, $ D\neq C$. Let $ X$ be a variable point on $ (BC)$, $ X\neq D$. Let $ Y$ be the second intersection point of $ AX$ and $ \omega$. Prove that the circumcircle of $ XYD$ passes through a fixed point.
2008 Moldova Team Selection Test
Find the number of even permutations of $ \{1,2,\ldots,n\}$ with no fixed points.
2008 Moldova Team Selection Test
Determine a subset $ A\subset \mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product.

Do not simply post the subset, show how you found it.
2008 Moldova Team Selection Test
Let $ p$ be a prime number and $ k,n$ positive integers so that $ \gcd(p,n)=1$. Prove that $ \binom{n\cdot p^k}{p^k}$ and $ p$ are coprime.
2008 Moldova Team Selection Test
In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC = \{M\}$ and $ O\cap CA = \{N\}$.

a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively.

b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.
2008 Moldova Team Selection Test
A non-empty set $ S$ of positive integers is said to be good if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S=\{a+1,a+2,a+3,\ldots,a+t\}$ is good, for any positive integer $ a$.



P.S.I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...
2007 Moldova Team Selection Test
Let $ABC$ be a triangle and $M,N,P$ be the midpoints of sides $BC, CA, AB$. The lines $AM, BN, CP$ meet the circumcircle of $ABC$ in the points $A_{1}, B_{1}, C_{1}$. Show that the area of triangle $ABC$ is at most the sum of areas of triangles $BCA_{1}, CAB_{1}, ABC_{1}$.
2007 Moldova Team Selection Test
Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.
2007 Moldova Team Selection Test
Consider a convex polygon $A_{1}A_{2}\ldots A_{n}$ and a point $M$ inside it. The lines $A_{i}M$ intersect the perimeter of the polygon second time in the points $B_{i}$. The polygon is called balanced if all sides of the polygon contain exactly one of points $B_{i}$ (strictly inside). Find all balanced polygons.



[Note: The problem originally asked for which $n$ all convex polygons of $n$ sides are balanced. A misunderstanding made this version of the problem appear at the contest]
2007 Moldova Team Selection Test
Find the least positive integers $m,k$ such that



a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube.



b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square.



The author is Vasile Suceveanu
2007 Moldova Team Selection Test
If $I$ is the incenter of a triangle $ABC$ and $R$ is the radius of its circumcircle then $$AI+BI+CI\leq 3R$$
2007 Moldova Team Selection Test
Let $M, N$ be points inside the angle $\angle BAC$ usch that $\angle MAB\equiv \angle NAC$. If $M_{1}, M_{2}$ and $N_{1}, N_{2}$ are the projections of $M$ and $N$ on $AB, AC$ respectively then prove that $M, N$ and $P$ the intersection of $M_{1}N_{2}$ with $N_{1}M_{2}$ are collinear.
2007 Moldova Team Selection Test
Consider five points in the plane, no three collinear. The convex hull of this points has area $S$. Prove that there exist three points of them that form a triangle with area at most $\frac{5-\sqrt 5}{10}S$
2007 Moldova Team Selection Test
Let $a_{1}, a_{2}, \ldots, a_{n}\in [0;1]$. If $S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}$ then prove that $$\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}$$
2007 Moldova Team Selection Test
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.
2007 Moldova Team Selection Test
Consider a triangle $ABC$, with corresponding sides $a,b,c$, inradius $r$ and circumradius $R$. If $r_{A}, r_{B}, r_{C}$ are the radii of the respective excircles of the triangle, show that

$$a^{2}\left(\frac 2{r_{A}}-\frac{r}{r_{B}r_{C}}\right)+b^{2}\left(\frac 2{r_{B}}-\frac{r}{r_{A}r_{C}}\right)+c^{2}\left(\frac 2{r_{C}}-\frac{r}{r_{A}r_{B}}\right)=4(R+3r) $$
2007 Moldova Team Selection Test
We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.
2007 Moldova Team Selection Test
Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
2007 Moldova Team Selection Test
If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial $$x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0$$ has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.
2007 Moldova Team Selection Test
Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that $$\angle{SBA}\equiv \angle{SCA}$$
2007 Moldova Team Selection Test
Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.
2006 Moldova Team Selection Test
Determine all even numbers $n$, $n \in \mathbb N$ such that $${ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, $$

where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.
2006 Moldova Team Selection Test
Consider a right-angled triangle $ABC$ with the hypothenuse $AB=1$. The bisector of $\angle{ACB}$ cuts the medians $BE$ and $AF$ at $P$ and $M$, respectively. If ${AF}\cap{BE}=\{P\}$, determine the maximum value of the area of $\triangle{MNP}$.
2006 Moldova Team Selection Test
Let $a,b,c$ be sides of the triangle. Prove that

$$ a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 . $$
2006 Moldova Team Selection Test
Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$

be the number of appearances of the number $n$ writing all of them on our $m$ circles.



a) Determine $r(n,m)$;



b) For $n=2006$, find the smallest $m$ for which $r(n,m)$ is a perfect square.



Example for half arc:    $1-1$;

$1-2-1$;

$1-3-2-3-1$;

$1-4-3-5-2-5-3-4-1$;

$1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$...
2006 Moldova Team Selection Test
Let $(a_n)$ be the Lucas sequence: $a_0=2,a_1=1, a_{n+1}=a_n+a_{n-1}$ for $n\geq 1$. Show that $a_{59}$ divides $(a_{30})^{59}-1$.
2006 Moldova Team Selection Test
Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.
2006 Moldova Team Selection Test
Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that



$a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$
2006 Moldova Team Selection Test
Let $A=\{1,2,\ldots,n\}$. Find the number of unordered triples $(X,Y,Z)$ that satisfy $X\bigcup Y \bigcup Z=A$
2006 Moldova Team Selection Test
Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.
2006 Moldova Team Selection Test
Let $n\in N$ $n\geq2$ and the set $X$ with $n+1$ elements. The ordered sequences $(a_{1}, a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots b_{n})$ of distinct elements of $X$ are said to be $\textit{separated}$ if there exists $i\neq j$ such that $a_{i}=b_{j}$. Determine the maximal number of ordered sequences of $n$ elements from $X$ such that any two of them are $\textit{separated}$.

Note: ordered means that, for example $(1,2,3)\neq(2,3,1)$.
2006 Moldova Team Selection Test
Positive real numbers $a,b,c$ satisfy the relation $abc=1$. Prove the inequality:

$\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3$.
2006 Moldova Team Selection Test
Let $f(n)$ denote the number of permutations $(a_{1}, a_{2}, \ldots ,a_{n})$ of the set $\{1,2,\ldots,n\}$, which satisfy the conditions: $a_{1}=1$ and $|a_{i}-a_{i+1}|\leq2$, for any $i=1,2,\ldots,n-1$. Prove that $f(2006)$ is divisible by 3.
2005 Moldova Team Selection Test
Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 + b^4 + c^4 = 3$. Prove that $ \sum\frac1{4 - ab}\leq1$, where the $ \sum$ sign stands for cyclic summation.



Alternative formulation: For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 + b^4 + c^4 = 3$, prove the inequality



$ \frac{1}{4-bc}+\frac{1}{4-ca}+\frac{1}{4-ab}\leq 1$.
2005 Moldova Team Selection Test
$$A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})$$

where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.
2005 Moldova Team Selection Test
$n$ is a positive integer, $K$ the set of polynoms of real variables $x_1,x_2,...,x_{n+1}$ and $y_1,y_2,...,y_{n+1}$, function $f:K\rightarrow K$ satisfies

$$f(p+q)=f(p)+f(q),\quad f(pq)=f(p)q+pf(q),\quad (\forall)p,q\in K.$$

If $f(x_i)=(n-1)x_i+y_i,\quad f(y_i)=2ny_i$ for all $i=1,2,...,n+1$ and

$$\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}$$

for any real $t$, prove, that for all $k=1,...,n+1$

$$f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}$$
2005 Moldova Team Selection Test
In triangle $ABC$, $M\in(BC)$, $\frac{BM}{BC}=\alpha$, $N\in(CA)$, $\frac{CN}{CA}=\beta$, $P\in(AB)$, $\frac{AP}{AB}=\gamma$.

Let $AM\cap BN=\{D\}$, $BN\cap CP=\{E\}$, $CP\cap AM=\{F\}$. Prove that

$S_{DEF}=S_{BMD}+S_{CNE}+S_{APF}$ iff $\alpha+\beta+\gamma=1$.
2005 Moldova Team Selection Test
Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72.

a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$.

b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.
2005 Moldova Team Selection Test
Does there exist such a configuration of 22 circles and 22 point, that any circle contains at leats 7 points and any point belongs at least to 7 circles?
2005 Moldova Team Selection Test
Find the largest positive $p$ ($p>1$) such, that $\forall a,b,c\in[\frac1p,p]$ the following inequality takes place

$$9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4$$
2005 Moldova Team Selection Test
Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that



$\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$,



where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.
2005 Moldova Team Selection Test
Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.
2003 Moldova Team Selection Test
Each side of an arbitrarly triangle is divided into $ 2002$ congruent segments. After that, each vertex is joined with all "division" points on the opposite side.

Prove that the number of the regions formed, in which the triangle is divided, is divisible by $ 6$.



Proposer: Dorian Croitoru
2003 Moldova Team Selection Test
The positive reals $ x,y$ and $ z$ are satisfying the relation $ x + y + z\geq 1$. Prove that:

$ \frac {x\sqrt {x}}{y + z} + \frac {y\sqrt {y}}{z + x} + \frac {z\sqrt {z}}{x + y}\geq \frac {\sqrt {3}}{2}$



Proposer: Baltag Valeriu
2003 Moldova Team Selection Test
Let $ ABCD$ be a quadrilateral inscribed in a circle of center $ O$. Let M and N be the midpoints of diagonals $ AC$ and $ BD$, respectively and let $ P$ be the intersection point of the diagonals $ AC$ and $ BD$ of the given quadrilateral .It is known that the points $ O,M,Np$ are distinct. Prove that the points $ O,N,A,C$ are concyclic if and only if the points $ O,M,B,D$ are concyclic.



Proposer: Dorian Croitoru
2003 Moldova Team Selection Test
Prove that the equation $ \frac {1}{a} + \frac {1}{b} + \frac {1}{c}+\frac{1}{abc} = \frac {12}{a + b + c}$ has infinitely many solutions $ (a,b,c)$ in natural numbers.
2003 Moldova Team Selection Test
Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form

$ P(X)=X^{2n}+(2n-10)X^{2n-1}+a_2X^{2n-2}+...+a_{2n-2}X^2+(2n-10)X+1$,

if it is known that all the roots of them are positive reals.



Proposer: Baltag Valeriu
2003 Moldova Team Selection Test
Consider the triangle $ ABC$ with side-lenghts equal to $ a,b,c$. Let $ p=\frac{a+b+c}{2}$, $ R$-the radius of circumcircle of the triangle $ ABC$, $ r$-the radius of the incircle of the triangle $ ABC$ and let $ l_a,l_b,l_c$ be the lenghts of bisectors drawn from $ A,B$ and $ C$, respectively, in the triangle $ ABC$. Prove that:

$ l_al_b+l_bl_c+l_cl_a\leq p\sqrt{3r^2+12Rr}$



Proposer: Baltag Valeriu
2003 Moldova Team Selection Test
The sides $ [AB]$ and $ [AC]$ of the triangle $ ABC$ are tangent to the incircle with center $ I$ of the $ \triangle ABC$ at the points $ M$ and $ N$, respectively. The internal bisectors of the $ \triangle ABC$ drawn form $ B$ and $ C$ intersect the line $ MN$ at the points $ P$ and $ Q$, respectively. Suppose that $ F$ is the intersection point of the lines $ CP$ and $ BQ$. Prove that $ FI\perp BC$.
2003 Moldova Team Selection Test
On the fields of a chesstable of dimensions $ n\times n$, where $ n\geq 4$ is a natural number, are being put coins. We shall consider a diagonal of table each diagonal formed by at least $ 2$ fields. What is the minimum number of coins put on the table, s.t. on each column, row and diagonal there is at least one coin? Explain your answer.
2003 Moldova Team Selection Test
Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called quadratic  iff at least one of the numbers $ a_1,a_1+a_2,...,a_1+a_2+a+...+a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.
2003 Moldova Team Selection Test
Let $ a_1,a_2,...,a_{2003}\geq 0$, such that $ a_1+a_2+...+a_{2003}=2$ and $ a_1a_2+a_2a_3+...+a_{2003}a_1=1$. Determine the minimum and maximum value of $ a_1^2+a_2^2+...+a_{2003}^2$.
2003 Moldova Team Selection Test
Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation:

$ (MA^2+MB^2+MC^2)^2=16(S_1^2+S_2^2+S_3^2)$
2003 Moldova Team Selection Test
A square-table of dimensions $ n\times n$, where $ n\in N^*$, is filled arbitrarly with the numbers $ 1,2,...,n^2$ such that every number appears on the table exactly one time. From each row of the table

is chosen the least number and then denote by $ x$ the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by $ y$ the biggest number from the numbers chosen. The table is called balanced iff $ x = y$. How many balanced tables we can obtain?
2002 Moldova Team Selection Test
Consider the triangular numbers $T_n = \frac{n(n+1)}{2} , n \in \mathbb N$.





(a) If $a_n$ is the last digit of $T_n$, show that the sequence $(a_n)$ is periodic and find its basic period.



(b) If $s_n$ is the sum of the first $n$ terms of the sequence $(T_n)$, prove that for every $n \geq 3$ there is at least one perfect square between $s_{n-1}$ and $s_n$.
2002 Moldova Team Selection Test
Prove that there exists a partition of the set $A = \{1^3, 2^3, \ldots , 2000^3\}$ into $19$ nonempty subsets such that the sum of elements of each subset is divisible by $2001^2$.