AT-OEMO-REG 2011

Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum $$\sum_{j=1}^{42}\frac{1}{p_j^2+1}$$ is not a unit fraction $\frac{1}{n^2}$ of some integer square number.

Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true: \begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128 \left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\text{.}\end{align*}

Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$ . Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$ . Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$ . Prove that $Q$ , $T$ , $U$ and $V$ are vertices of a trapezoid.

Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that $$\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\text{.}$$ Determine the set of elements of $(a_n)$ .