Let $ABC$ be a non-right isosceles triangle such that $AC = BC$. Let $D$ be such a point on the perpendicular bisector of $AB$, that $AD$ is tangent on the $ABC$ circumcircle. Let $E$ be such a point on $AB$, that $CE$ and $AD$ are perpendicular and let $F$ be the second intersection of line $AC$ and the circle $CDE$. Prove that $DF$ and $AB$ are parallel.

Prove, that for any positive real numbers $a, b, c$ who satisfy $a^2+b^2+c^2=1$ the following inequality holds. $\sqrt{\frac{1}{a}-a}+\sqrt{\frac{1}{b}-b}+\sqrt{\frac{1}{c}-c} \geq \sqrt{2a}+\sqrt{2b}+\sqrt{2c}$

Let $n$ be any positive integer and $M$ a set that contains $n$ positive integers. A sequence with $2^n$ elements is christmassy if every element of the sequence is an element of $M$. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.

Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.

Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$. Prove that, $PQ$ bisects $BC$.

Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.

Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$. Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.