JBMO 2015

Find all prime numbers $a, b, c$ and positive integers $k$ which satisfy the equation

$$a^{2}+b^{2}+16 \cdot c^{2}=9 \cdot k^{2}+1$$

Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression

$$A=\frac{2-a^{3}}{a}+\frac{2-b^{3}}{b}+\frac{2-c^{3}}{c}$$

$19^{\text {th }}$ Junior Balkan Mathematical Olympiad June 24-29, 2015, Belgrade, Serbia

Let $\triangle A B C$ be an acute triangle. The lines $l_{1}, l_{2}$ are perpendicular to $A B$ at the points $A, B$ respectively. The perpendicular lines from the midpoint $M$ of $A B$ to the sides of the triangle $A C, B C$ intersect the lines $l_{1}, l_{2}$ at the points $E, F$, respectively. If $D$ is the intersection point of $E F$ and $M C$, prove that

$$\angle A D B=\angle E M F$$

An $L$-figure is one of the following four pieces, each consisting of three unit squares:

A $5 \times 5$ board, consisting of 25 unit squares, a positive integer $k \leq 25$ and an unlimited supply $L$-figures are given. Two players, $\boldsymbol{A}$ and $\boldsymbol{B}$, play the following game: starting with $\boldsymbol{A}$ they alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares.

We say that a placement of $L$-figures on unmarked unit squares is called good if the $L$-figure do not overlap and each of them covers exactly three unmarked unit squares of the board. $\boldsymbol{B}$ wins if every $\boldsymbol{g o o d}$ placement of $L$-figures leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which $\boldsymbol{B}$ has a winning strategy.