ELMO 2012

In acute triangle $ABC$ , let $D,E,F$ denote the feet of the altitudes from $A,B,C$ , respectively, and let $\omega$ be the circumcircle of $\triangle AEF$ . Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$ , respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$ .

Ray Li.

Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ( $k\ge1$ ) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$ , exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$ .

David Yang.

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$ . Show that there exists a positive integer $n$ such that $a_n=b_n$ .

David Yang.

Let $ABC$ be an acute triangle with $AB<AC$ , and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$ . Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$ . Prove that $\angle PBA=\angle PCA$ .

Calvin Deng.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$ . The dials are initially set to states $d_1, d_2, \ldots, d_n$ , where $0\le d_i\le c-1$ for each $1\le i\le n$ . Unfortunately, the actual states of the dials (the $d_i$ 's) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ( $0\le c_i\le c-1$ ), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$ . After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$ 's by $k$ (so that for every $i$ , $d_i$ is replaced by $d_{i-k}$ , where indices are taken modulo $n$ ).

Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime.

Bobby Shen.