For every positive integer $n$ we define $a_{n}$ as the last digit of the sum $1+2+\cdots+n$. Compute $a_{1}+a_{2}+\cdots+a_{1992}$.

Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$.

Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle. a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5. b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.

Let $\{a_{n}\}_{n \geq 0}$ and $\{b_{n}\}_{n \geq 0}$ be two sequences of integer numbers such that: i. $a_{0}=0$, $b_{0}=8$. ii. For every $n \geq 0$, $a_{n+2}=2a_{n+1}-a_{n}+2$, $b_{n+2}=2b_{n+1}-b_{n}$. iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for every $n \geq 0$. Find at least two values of the pair $(a_{1992},\, b_{1992})$.

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.