OPM 2011

A set of $n$ lights, numbered $1$ to $n$ , are initially off. At every moment, it is possible to perform one of the following operations: $\bullet$ change the state of lamp $1$ , $\bullet$ change the state of lamp $2$ , as long as lamp $1$ is on, $\bullet$ change the state of lamp $k > 2$ , as long as lamp $k - 1$ is on and all lamps $1, . . . , k - 2$ are off. It shows that it is possible, after a certain number of operations, to have only the lamp left on.

In a class of $14$ boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers $0, 1, 2, 3, 4, 5$ and $6$ . Proves that there are two colleagues with the same first name and the same last name

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$ .

The number $1000$ can be written as the sum of $16$ consecutive natural numbers: $$1000 = 55 + 56 + ... + 70.$$ Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .