OIM 1992

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 $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$ .