KURSCHAK 2010

We have $n$ keys, each of them belonging to exactly one of $n$ locked chests. Our goal is to decide which key opens which chest. In one try we may choose a key and a chest, and check whether the chest can be opened with the key. Find the minimal number $p(n)$ with the property that using $p(n)$ tries, we can surely discover which key belongs to which chest.

Consider a triangle $ABC$ , with the points $A_1$ , $A_2$ on side $BC$ , $B_1,B_2\in\overline{AC}$ , $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$ , $BA_1<BA_2$ , $CB_1<CB_2$ . Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$ . Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$ , let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$ . Prove that the lines $AA^*$ , $BB^*$ , $CC^*$ are concurrent.

For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ( $1\le i\le n,1\le j\le k$ ) give pairwise different residues modulo $nk$ ?