Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$

In a school there $b$ teachers and $c$ students. Suppose that a) each teacher teaches exactly $k$ students, and b)for any two (distinct) students , exactly $h$ teachers teach both of them. Prove that $\frac{b}{h}=\frac{c(c-1)}{k(k-1)}$.

Points $P$ and $Q$ are taken sides $AB$ and $AC$ of a triangle $ABC$ respectively such that $\hat{APC}=\hat{AQB}=45^{0}$. The line through $P$ perpendicular to $AB$ intersects $BQ$ at $S$, and the line through $Q$ perpendicular to $AC$ intersects $CP$ at $R$. Let $D$ be the foot of the altitude of triangle $ABC$ from $A$. Prove that $SR\parallel BC$ and $PS,AD,QR$ are concurrent.

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.