CWMO 2013

Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$ are perfect squares?

Let the integer $n \ge 2$ , and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right]$ .Prove that$$\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.$$

Let $ABC$ be a triangle, and $B_1,C_1$ be its excenters opposite $B,C$ . $B_2,C_2$ are reflections of $B_1,C_1$ across midpoints of $AC,AB$ . Let $D$ be the extouch at $BC$ . Show that $AD$ is perpendicular to $B_2C_2$

There are $n$ coins in a row, $n\geq 2$ . If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$ . No move is allowed if all $n$ coins are tails. Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $\lfloor\frac {2^m}{3}\rfloor$ moves

A nonempty set $A$ is called an $n$ -level-goodset if $A \subseteq \{1,2,3,\ldots,n\}$ and $|A| \le \min_{x\in A} x$ (where $|A|$ denotes the number of elements in $A$ and $\min_{x\in A} x$ denotes the minimum of the elements in $A$ ). Let $a_n$ be the number of $n$ -level-good sets. Prove that for all positive integers $n$ we have $a_{n+2}=a_{n+1}+a_{n}+1$ .

Let $PA, PB$ be tangents to a circle centered at $O$ , and $C$ a point on the minor arc $AB$ . The perpendicular from $C$ to $PC$ intersects internal angle bisectors of $AOC,BOC$ at $D,E$ . Show that $CD=CE$

Label sides of a regular $n$ -gon in clockwise direction in order 1,2,..,n. Determine all integers n ( $n\geq 4$ ) satisfying the following conditions: (1) $n-3$ non-intersecting diagonals in the $n$ -gon are selected, which subdivide the $n$ -gon into $n-2$ non-overlapping triangles; (2) each of the chosen $n-3$ diagonals are labeled with an integer, such that the sum of labeled numbers on three sides of each triangles in (1) is equal to the others;

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$ .