SMO 2012 Senior Round 2

A circle $\omega$ through the incentre $I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of $BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.

Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

If $46$ squares are colored red in a $9\times9$ board, show that there is a $2\times2$ block on the board in which at least $3$ of the squares are colored red.

Let $a_0,a_1,a_2,\ldots,a_n,a_{n+1}$ be a finite sequence of real numbers satisfying
$$a_0=a_{n+1}=0,$$
$$a_{k-1}-2a_k+a_{k+1}\le1\qquad(k=1,2,\ldots,n).$$
Prove that for $k=0,1,\ldots,n+1$,
$$a_k\le\frac{k(n+1-k)}2.$$

Prove that for any real numbers $a,b,c,d\ge0$ with $a+b=c+d=2$,
$$(a^2+c^2)(a^2+d^2)(b^2+c^2)(b^2+d^2)\le25.$$