In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

The sequences $(a_{n})$, $(b_{n})$ are defined by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$?

In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?

A convex $ABCDE$ is inscribed in a unit circle, $AE$ being its diameter. If $AB = a$, $BC = b$, $CD = c$, $DE = d$ and $ab = cd =\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d.$

Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$

If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, find the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$.