Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Romania Team Selection Test
Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incentre of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$.



Radu Gologan
2010 Romania Team Selection Test
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)



Vlad Matei
2010 Romania Team Selection Test
Let  $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set  $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements.



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2010 Romania Team Selection Test
Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point.



Mihai Chis
2010 Romania Team Selection Test
Let $ABC$ be a scalene triangle. The tangents at the perpendicular foot dropped from $A$ on the line $BC$ and the midpoint of the side $BC$ to the nine-point circle meet at the point $A'$\,; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent.



Gazeta Matematica
2010 Romania Team Selection Test
Let  $\mathcal{L}$ be a finite collection of  lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$.



B. Aronov et al.
2010 Romania Team Selection Test
Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set.



Vasile Pop
2010 Romania Team Selection Test
Let $\ell$ be a line, and let  $\gamma$ and $\gamma'$ be two circles. The line  $\ell$ meets $\gamma$ at points $A$ and $B$, and  $\gamma'$ at points $A'$ and $B'$. The tangents to $\gamma$ at $A$ and  $B$ meet at point  $C$, and the tangents to $\gamma'$ at  $A'$ and  $B'$ meet at point  $C'$. The lines  $\ell$ and  $CC'$ meet at point $P$. Let  $\lambda$ be a variable line through  $P$ and let  $X$ be one of the points where  $\lambda$ meets $\gamma$, and $X'$ be one of the points where  $\lambda$ meets $\gamma'$. Prove that the point of intersection of the lines  $CX$ and $C'X'$ lies on a fixed circle.



Gazeta Matematica
2010 Romania Team Selection Test
Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let  $d$ be the greatest common divisor of the numbers  $n_1, n_2, \ldots, n_p$. Prove that the polynomial

$$\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}$$

is irreducible in $\mathbb{Q}[X]$.



Beniamin Bogosel
2010 Romania Team Selection Test
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.



AMM Magazine
2010 Romania Team Selection Test
Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$.



Cezar Lupu & Vlad Matei
2010 Romania Team Selection Test
Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that

$$4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. $$

Determine the maximum of $N(S)$ when $S$ runs through all $n$-element sets of vectors in the plane.



***
2009 Romania Team Selection Test
For non-empty subsets $A,B \subset \mathbb{Z}$ define $$A+B=\{a+b:a\in A, b\in B\},\ A-B=\{a-b:a\in A, b\in B\}.$$



In the sequel we work with non-empty finite subsets of $\mathbb{Z}$.



Prove that we can cover $B$ by at most $\frac{|A+B|}{|A|}$ translates of $A-A$, i.e. there exists $X\subset Z$ with $|X|\leq \frac{|A+B|}{|A|}$ such that $$B\subseteq \cup_{x\in X} (x+(A-A))=X+A-A.$$
2009 Romania Team Selection Test
Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.
2009 Romania Team Selection Test
Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with lamp $k+1$,..., lamp $n-1$ with lamp $n$, lamp $n$ with lamp $1$. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.
2009 Romania Team Selection Test
We call Golomb ruler a ruler of length $l$, bearing $k+1\geq 2$ marks $0<a_1<\ldots <a_{k-1}<l$, such that the lengths that can be measured using marks on the ruler are consecutive integers starting with $1$, and each such length be measurable between just two of the gradations of the ruler. Find all Golomb rulers.
2009 Romania Team Selection Test
A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.
2009 Romania Team Selection Test
Prove that pentagon $ ABCDE$ is cyclic if and only if

$$\mathrm{d(}E,AB\mathrm{)}\cdot \mathrm{d(}E,CD\mathrm{)} = \mathrm{d(}E,AC\mathrm{)}\cdot \mathrm{d(}E,BD\mathrm{)} = \mathrm{d(}E,AD\mathrm{)}\cdot \mathrm{d(}E,BC\mathrm{)}$$

where $ \mathrm{d(}X,YZ\mathrm{)}$ denotes the distance from point $ X$ ot the line $ YZ$.
2009 Romania Team Selection Test
Let $ABCD$ be a circumscribed quadrilateral such that $AD>\max\{AB,BC,CD\}$, $M$ be the common point of $AB$ and $CD$ and $N$ be the common point of $AC$ and $BD$. Show that $$90^{\circ}<m(\angle AND)<90^{\circ}+\frac{1}{2}m(\angle AMD).$$



Fixed, thank you Luis.
2009 Romania Team Selection Test
Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.
2009 Romania Team Selection Test
Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.
2009 Romania Team Selection Test
Given an integer $n\geq 2$, determine the maximum value the sum $x_1+\cdots+x_n$ may achieve, as the $x_i$ run through the positive integers, subject to $x_1\leq x_2\leq \cdots \leq x_n$ and $x_1+\cdots+x_n=x_1 x_2\cdots x_n$.
2009 Romania Team Selection Test
Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that $$|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.$$ Prove that $$\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}$$ and find the cases of equality.
2009 Romania Team Selection Test
Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.
2009 Romania Team Selection Test
Let $a$ and $n$ be two integers greater than $1$. Prove that if $n$ divides $(a-1)^k$ for some integer $k\geq 2$, then $n$ also divides $a^{n-1}+a^{n-2}+\cdots+a+1$.
2009 Romania Team Selection Test
Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent:

a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$;

b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$
2009 Romania Team Selection Test
Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$.
2009 Romania Team Selection Test
Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.
2009 Romania Team Selection Test
Prove that the edges of a finite simple planar graph (with no loops, multiple edges) may be oriented in such a way that at most three fourths of the total number of dges of any cycle share the same orientation. Moreover, show that this is the best global bound possible.



Comment: The actual problem in the TST asked to prove that the edges can be $2$-colored so that the same conclusion holds. Under this circumstances, the problem is wrong and a counterexample was found in the contest by Marius Tiba.
2009 Romania Team Selection Test
Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.
2008 Romania Team Selection Test
Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n+1$.
2008 Romania Team Selection Test
Let $ a_i, b_i$ be positive real numbers, $ i=1,2,\ldots,n$, $ n\geq 2$, such that $ a_i<b_i$, for all $ i$, and also $$ b_1+b_2+\cdots + b_n < 1 + a_1+\cdots + a_n.$$ Prove that there exists a $ c\in\mathbb R$ such that for all $ i=1,2,\ldots,n$, and $ k\in\mathbb Z$ we have $$ (a_i+c+k)(b_i+c+k) > 0.$$
2008 Romania Team Selection Test
Let $ ABCDEF$ be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles $ ACE$ or $ BDF$ is at least 1.
2008 Romania Team Selection Test
Prove that there exists a set $ S$ of $ n - 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.
2008 Romania Team Selection Test
Find the greatest common divisor of the numbers $$ 2^{561}-2, 3^{561}-3, \ldots, 561^{561}-561.$$
2008 Romania Team Selection Test
Let $ n \geq 3$ be an odd integer. Determine the maximum value of

$$ \sqrt{|x_{1}-x_{2}|}+\sqrt{|x_{2}-x_{3}|}+\ldots+\sqrt{|x_{n-1}-x_{n}|}+\sqrt{|x_{n}-x_{1}|},$$

where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.
2008 Romania Team Selection Test
Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} + n\ |\ k = 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?
2008 Romania Team Selection Test
Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side.



Note. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.
2008 Romania Team Selection Test
Let $ G$ be a connected graph with $ n$ vertices and $ m$ edges such that each edge is contained in at least one triangle. Find the minimum value of $ m$.
2008 Romania Team Selection Test
Let $ ABC$ be a triangle with $ \measuredangle{BAC} < \measuredangle{ACB}$. Let $ D$, $ E$ be points on the sides $ AC$ and $ AB$, such that the angles $ ACB$ and $ BED$ are congruent. If $ F$ lies in the interior of the quadrilateral $ BCDE$ such that the circumcircle of triangle $ BCF$ is tangent to the circumcircle of $ DEF$ and the circumcircle of $ BEF$ is tangent to the circumcircle of $ CDF$, prove that the points $ A$, $ C$, $ E$, $ F$ are concyclic.



Author: Cosmin Pohoata
2008 Romania Team Selection Test
Let $ ABC$ be an acute triangle with orthocenter $ H$ and let $ X$ be an arbitrary point in its plane. The circle with diameter $ HX$ intersects the lines $ AH$ and $ AX$ at $ A_{1}$ and $ A_{2}$, respectively. Similarly, define $ B_{1}$, $ B_{2}$, $ C_{1}$, $ C_{2}$. Prove that the lines $ A_{1}A_{2}$, $ B_{1}B_{2}$, $ C_{1}C_{2}$ are concurrent.



Click to reveal hidden textRemark. The triangle obviously doesn't need to be acute.
2008 Romania Team Selection Test
Let $ m,\ n \geq 3$ be positive odd integers. Prove that $ 2^{m}-1$ doesn't divide $ 3^{n}-1$.
2008 Romania Team Selection Test
Let $ n$ be a nonzero positive integer. A set of persons is called a $ n$-balanced set if in any subset of $ 3$ persons there exists at least two which know each other and in each subset of $ n$ persons there are two which don't know each other. Prove that a $ n$-balanced set has at most $ (n - 1)(n + 2)/2$ persons.
2008 Romania Team Selection Test
Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.
2008 Romania Team Selection Test
Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of  $ \{1, 2, ..., m + n - 1\}$ such that $ |A| = m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.
2008 Romania Team Selection Test
Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n-1}+1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j = A_k$.
2008 Romania Team Selection Test
Let $ n$ be a nonzero positive integer. Find $ n$ such that there exists a permutation $ \sigma \in S_{n}$ such that

$$ \left| \{ |\sigma(k) - k| \ : \ k \in \overline{1, n} \}\right | = n.$$
2008 Romania Team Selection Test
Let $ ABC$ be a triangle and let $ \mathcal{M}_{a}$, $ \mathcal{M}_{b}$, $ \mathcal{M}_{c}$ be the circles having as diameters the medians $ m_{a}$, $ m_{b}$, $ m_{c}$ of triangle $ ABC$, respectively. If two of these three circles are tangent to the incircle of $ ABC$, prove that the third is tangent as well.
2008 Romania Team Selection Test
Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that

$$ 3 \cdot 2^{n-1} - 2 \le f(n) \le 3^n - 2.$$
2007 Romania Team Selection Test
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that $$a_{1}^{2}+\cdots+a_{n}^{2}=1,$$

then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.
2007 Romania Team Selection Test
Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that $$|f(x)-f(y)|\leq (x-y)^{2}$$ for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.
2007 Romania Team Selection Test
Let $A_{1}A_{2}\ldots A_{2n}$ be a convex polygon and let $P$ be a point in its interior such that it doesn't lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines $PA_{1}$, $\ldots$, $PA_{2n}$ intersects it in its interior.
2007 Romania Team Selection Test
Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.
2007 Romania Team Selection Test
Let

$$f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}$$

be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even.

Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd.

Prove that $f$ has an integer root.
2007 Romania Team Selection Test
Let $ABC$ be a triangle, $E$ and $F$ the points where the incircle and $A$-excircle touch $AB$, and $D$ the point on $BC$ such that the triangles $ABD$ and $ACD$ have equal in-radii. The lines $DB$ and $DE$ intersect the circumcircle of triangle $ADF$ again in the points $X$ and $Y$.



Prove that $XY\parallel AB$ if and only if $AB=AC$.
2007 Romania Team Selection Test
Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$.



Dan Schwarz
2007 Romania Team Selection Test
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples $$\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) $$ such that

$$\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. $$

(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.

(b) Compute $M(3)$ and $M(4)$.
2007 Romania Team Selection Test
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find

$$\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. $$
2007 Romania Team Selection Test
Prove that for $n, p$ integers, $n \geq 4$ and $p \geq 4$, the proposition $\mathcal{P}(n, p)$

$$\sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}\quad \textrm{for}\quad x_{i}\in \mathbb{R}, \quad x_{i}> 0 , \quad i=1,\ldots,n \ ,\quad \sum_{i=1}^{n}x_{i}= n,$$ is false.



Dan Schwarz



RemarkIn the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions $\mathcal{P}(4, 3)$ are $\mathcal{P}(3, 4)$ true.
2007 Romania Team Selection Test
Let $a_{i}$, $i = 1,2, \dots ,n$, $n \geq 3$, be positive integers, having the greatest common divisor 1, such that $$a_{j}\textrm{ divide }\sum_{i = 1}^{n}a_{i}$$

for all $j = 1,2, \dots ,n$. Prove that $$\prod_{i = 1}^{n}a_{i}\textrm{ divides }\Big{(}\sum_{i = 1}^{n}a_{i}\Big{)}^{n-2}.$$
2007 Romania Team Selection Test
The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with  $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.
2007 Romania Team Selection Test
Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.
2007 Romania Team Selection Test
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that

$$ [A_{1}A_{2}A_{3}] = [A_{2}A_{3}A_{4}] = [A_{3}A_{4}A_{5}] = [A_{4}A_{5}A_{1}] = [A_{5}A_{1}A_{2}].$$

Prove that there exists a point  $ M$ in the plane of the pentagon such that

$$ [A_{1}MA_{2}] = [A_{2}MA_{3}] = [A_{3}MA_{4}] = [A_{4}MA_{5}] = [A_{5}MA_{1}].$$

Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.
2007 Romania Team Selection Test
Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element  $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if $$c > n \left( \frac{2}{3}\right )^{k+1},$$ where  $k$ is the exponent of $2$ in the factoring of $c$.
2007 Romania Team Selection Test
i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$,

the $p$-th term of the progression is also prime.



ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime.



Dan Schwarz
2007 Romania Team Selection Test
In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.
2007 Romania Team Selection Test
Let $ABC$ be a triangle, and $\omega_{a}$, $\omega_{b}$, $\omega_{c}$ be circles inside $ABC$, that are tangent (externally) one to each other, such that $\omega_{a}$ is tangent to $AB$ and $AC$, $\omega_{b}$ is tangent to $BA$ and $BC$, and $\omega_{c}$ is tangent to $CA$ and $CB$. Let $D$ be the common point of $\omega_{b}$ and $\omega_{c}$, $E$ the  common point of $\omega_{c}$ and $\omega_{a}$, and $F$ the common point of $\omega_{a}$ and $\omega_{b}$. Show that the lines $AD$, $BE$ and $CF$ have a common point.
2007 Romania Team Selection Test
Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$.



a) If $\angle B=135^{\circ}$, find the length of $[BD]$.



b) Find the minimum of the length of $[BD]$.
2007 Romania Team Selection Test
Let $ ABCD$ be a parallelogram with no angle equal to $ 60^{\textrm{o}}$. Find all pairs of points $ E, F$, in the plane of $ ABCD$, such that triangles $ AEB$ and $ BFC$ are isosceles, of basis $ AB$, respectively $ BC$, and triangle $ DEF$ is equilateral.



Valentin Vornicu
2007 Romania Team Selection Test
The world-renowned Marxist theorist Joric is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the defect of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute $$\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.$$

Iurie Boreico