Prove that there exists infinetly many natural number $n$ such that at least one of the numbers $2^{2^n}+1$ and $2018^{2^n}+1$ is not a prime.

Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that $$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$

Ana and Bob are playing the following game. First, Bob draws triangle $ABC$ and a point $P$ inside it. Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$. Finally, Ana draw a triangle $V_1V_2V_3$. For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$ to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.

An isosceles trapezium is called right if only one pair of its sides are parallel (i.e parallelograms are not right). A dissection of a rectangle into $n$ (can be different shapes) right isosceles trapeziums is called strict if the union of any $i,(2\leq i \leq n)$ trapeziums in the dissection do not form a right isosceles trapezium. Prove that for any $n, n\geq 9$ there is a strict dissection of a $2017 \times 2018$ rectangle into $n$ right isosceles trapeziums. Proposed by Bojan Basic

Let $H $ be the orthocenter of $ABC $ ,$AB\neq AC $ ,and let $F $ be a point on circumcircle of $ABC $ such that $\angle AFH=90^{\circ} $.$K $ is the symmetric point of $H $ wrt $B $.Let $P $ be a point such that $\angle PHB=\angle PBC=90^{\circ} $,and $Q $ is the foot of $B $ to $CP $.Prove that $HQ $ is tangent to tge circumcircle of $FHK $.

For any positive integer $n$, define $$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?