BALKAN-SL 2017

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that

$$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}} \leq 1 .$$

Consider the sequence of rational numbers defned by $x_{1}=\frac{4}{3}$ and $x_{n+1}=\frac{x_{n}^{2}}{x_{n}^{2}-x_{n}+1}, n \geq 1$. Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n} x_{k}$ is a perfect square.

Find all the functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that:

$$n+f(m) \mid f(n)+n f(m)$$

for any $m, n \in \mathbb{N}$

Let $M=\left\{(a, b, c) \in \mathbb{R}^{3}: 0<a, b, c<\frac{1}{2}\right.$ with $\left.a+b+c=1\right\}$ and $f: M \rightarrow \mathbb{R}$ given as

$$f(a, b, c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{a b c}$$

Find the best (real) bounds $\alpha$ and $\beta$ such that

$$f(M)=\{f(a, b, c):(a, b, c) \in M\} \subseteq[\alpha, \beta]$$

and determine whether any of them is achievable.

Consider integers $m \geq 2$ and $n \geq 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0), P(1), \ldots, P(n)$ are all perfect powers of $m$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying

$$f\left(x+y f\left(x^{2}\right)\right)=f(x)+x f(x y)$$

for all real numbers $x$ and $y$.

A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?

Let $n, a, b, c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

Let $n \geq 4$ points in the plane, no three of them are collinear. Prove that the number of parallelograms of area 1 , formed by these points, is at most $\frac{n^{2}-3 n}{4}$.

For any set of points $A_{1}, A_{2}, \ldots, A_{n}$ on the plane, one defines $r\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \geq 3$, there are indices $i, j, k$ such that

$$r\left(A_{1}, A_{2}, \ldots, A_{n}\right)=r\left(A_{1}, A_{,}, A_{k}\right)$$

What is the least positive integer $k$ such that, in every convex 101 -gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

Let $A B C$ be an acute triangle. Variable points $E$ and $F$ are on sides $A C$ and $A B$ respectively such that $B C^{2}=B A \cdot B F+C E \cdot C A$. As $E$ and $F$ vary prove that the circumcircle of $A E F$ passes through a fixed point other than $A$.

Let $A B C$ be an acute triangle and $D$ a variable point on side $A C$. Point $E$ is on $B D$ such that $B E=\frac{B C^{2}-C D \cdot C A}{B D}$. As $D$ varies on side $A C$ prove that the circumcircle of $A D E$ passes through a fixed point other than $A$.

Let $A B C$ be a triangle with $A B<A C$ inscribed into a circle $c$. The tangent of $c$ at the point $C$ meets the parallel from $B$ to $A C$ at the point $D$. The tangent of $c$ at the point $B$ meets the parallel from $C$ to $A B$ at the point $E$ and the tangent of $c$ at the point $C$ at the point $L$. Suppose that the circumcircle $c_{1}$ of the triangle $B D C$ meets $A C$ at the point $T$ and the circumcircle $c_{2}$ of the triangle $B E C$ meets $A B$ at the point $S$. Prove that the lines $S T, B C, A L$ are concurrent.

The acuteangled triangle $A B C$ with circumcenter $O$ is given. The midpoints of the sides $B C$, $C A$ and $A B$ are $D, E$ and $F$ respectivelly. An arbitrary point $M$ on the side $B C$, different of $D$, is choosen. The straight lines $A M$ and $E F$ intersects at the point $N$ and the straight line $O N$ cut again the circumscribed circle of the triangle $O D M$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $D P$ belongs on the nine points circle of the triangle $A B C$.

Let $A B C$ be an acute angled triangle with ortocenter $H$, centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $A H$ and $A G$ with side $B C$. Rays $M H$ and $D G$ interect $\omega$ again at $P$ and $Q$ respectively. Prove that $P D$ and $Q M$ intersect on $\omega$.

Construct outside the acute-angled triangle $A B C$ the isosceles triangles $A B A_{B}, A B B_{A}$, $A C A_{C}, A C C_{A}, B C B_{C}$ and $B C C_{B}$, so that

$$A B=A B_{A}=B A_{B}, A C=A C_{A}=C A_{C}, B C=B C_{B}=C B_{C}$$

and

$$\measuredangle B A B_{A}=\measuredangle A B A_{B}=\measuredangle C A C_{A}=\measuredangle A C A_{C}=\measuredangle B C B_{C}=\measuredangle C B C_{B}=\alpha<90^{\circ} .$$

Prove that the perpendiculars from $A$ to $B_{A} C_{A}$, from $B$ to $A_{B} C_{B}$ and from $C$ to $A_{C} B_{C}$ are concurrent.

Let $A B C$ be an acute triangle with $A B \neq A C$ and circumcirle $\Gamma$. The angle bisector of $B A C$ intersects $B C$ and $\Gamma$ at $D$ and $E$ respectively. Circle with diameter $D E$ intersects $\Gamma$ again at $F \neq E$. Point $P$ is on $A F$ such that $P B=P C$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $A B$ and $A C$ respectively. Let $H$ and $H^{\prime}$ be the ortocenters of $A B C$ and $A X Y$ respectively. $A H$ meets $\Gamma$ again at $Q$. If $A H^{\prime}$ and $H H^{\prime}$ intersect the circle with diameter $A H$ again at points $S$ and $T$, respectively, prove that the lines $A T, H S$ and $F Q$ are concurrent.

Given an acute triangle $\triangle A B C(A C \neq A B)$ and let $(C)$ be its circumcircle. The excircle $\left(C_{1}\right)$ corresponding to the vertex $A$, of center $I_{a}$, tangents to the side $B C$ at the point $D$ and to the extensions of the sides $A B, A C$ at the points $E, Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $\left(C_{1}\right), H$ the orthocenter of the triangle $\triangle E D Z$ and $N$ the midpoint of segment $E Z$. The parallel line through the point $I_{a}$ to the line $H L$ meets the line $H I$ at the point $G$. Prove that the perpendicular line ( $e$ ) through the point $N$ to the to the line $B C$ and the parallel line $(\delta)$ through the point $G$ to the line $I L$ meet each other on the line $\mathrm{HI}_{a}$