IL-GILLIS 2015

[list=a] [] Find an example of three positive integers $a,b,c$ satisfying $31a+30b+28c=365$ . [] Prove that any triplet $a,b,c$ satisfying the above condition, also satisfies $a+b+c=12$ . [/list]

A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$ . Evaluate the triangle's area.

Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.

Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$ , and $m^k$ is divisible by $k^m$ . [list=a] [] Prove that $n^k$ is divisible by $k^n$ . [] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1. [/list]

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$ , tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$ . Let $K$ be the tangency point of $S_1$ to the face $ABC$ , and let $L$ be the tangency point of $S_2$ to the face $ABC$ . Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$ .

Prove that $L,T,K$ lie on one line.

Let $n\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp is on at minute $t+1$ if and only if at minute $t$ , exactly one of its neighbors is on (the two lamps at the ends have one neighbor each, all other lamps have two neighbors).

For which values of $n$ can we guarantee that all lamps will be off after some time?

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$ . Let $p\geq3$ be a prime number.

[list=a] [] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$ . [] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$ . [/list]