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[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2015

  1. Let $ABC$ be an acute triangle with $AB \neq AC$. Also let $M$ be the midpoint of the side $BC$, $H$ the orthocenter of the triangle $ABC$, $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$. Prove that $O_1AMO_2$ is a parallelogram.
  2. Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube, there will be a vertex of the cube which has the three vertices adjacent to it colored in red.
  3. Let $x,y,z>0$. Show that $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$
  4. Solve in nonnegative integers the following equation $$21^x+4^y=z^2$$
  5. Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel. Denote by $O$ the intersection of the diagonals, $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$. Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$. Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$.
  6. Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ (for example $12$ works because $1 \cdot 2$ divides $12$).
  7. Let $a,b,c>0$ such that $a \geq bc^2$, $b \geq ca^2$ and $c \geq ab^2$. Find the maximum value that the expression $$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.
  8. Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio?.
  9. Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of $ \angle ABC$ with $D \in AC$. Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$. The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$. Denote by $J$ the symmetric of $A$ with respect to $K$. Prove that if $DJ$ intersects $AM$ in $O$ the $J$, $B$, $M$, $O$ are concyclic.
  10. Define the set $M_q=\{x \in \mathbb{Q} \mid x^3-2015x=q \}$, where $q$ is an arbitrary rational number.
    a) Show that there exists values for $q$ such that the set is null as well as values for which it has exactly one element.
    b) Determine all the possible values for the cardinality of $M_q$.
  11. Find all the triplets of real numbers $(x , y , z)$ such that $$\begin{cases}y&=\dfrac{x^3+12x}{3x^2+4},\\ z&=\dfrac{y^3+12y}{3y^2+4},\\ x&=\dfrac{z^3+12z}{3z^2+4}\end{cases}$$
  12. Let $ABC$ be an acute triangle with $AB \neq AC$ and denote its orthocenter by $H$. The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$, respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.
  13. The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$. A move consists in choosing three consecutive vertices and changing the signs from the vertices, from $+$ to $-$ and from $-$ to $+$.
    a) Prove that if $n=2015$ then for any initial configuration of signs, there exists a sequence of moves such that we'll arrive at a configuration with only $+$ signs.
    b) Prove that if $n=2016$, then there exists an initial configuration of signs such that no matter how we make the moves we'll never arrive at a configuration with only $+$ signs.
  14. Let $n\in \mathbb{N}$, $n \geq 4$. Determine all sets $$A = \{a_1, a_2, . . . , a_n\} \subset \mathbb{N}$$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$
  15. Solve in $\mathbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$
  16. Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$
  17. We have $n$ integers $a_1, a_2,. . . , a_n$, not necessarily distinct, with sum $2S.$ An integer $k$ is called separator if $k$ of the numbers can be chosen with sum equal to $S.$ What is the maximum possible number of separators?
  18. Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
  19. Two players, $A$ and $B,$ alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and in his first move he must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as his opponent took in the previous move. The player who takes the last stone wins. Which player has a winning strategy?
  20. Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$
  21. Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP$, $BP$ and $CP$ intersect circle $\omega$ for the second time at $D$, $E$ and $F$, respectively. If $A'$, $B'$, $C'$ are the reflections of $A$, $B$, $C$ with respect to the lines $EF$, $FD$, $DE$, respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.

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MOlympiad: [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2015
[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2015
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