SMO 2014 Junior Round 2

Consider the integers formed using the digits 0, 1, 2, 3, 4, 5, 6, without repetition. Find the largest multiple of 55. Justify your answer.

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.

In the triangle $ABC$, the bisector of $\angle A$ intersects the bisector of $\angle B$ at the point $I$; $D$ is the foot of the perpendicular from $I$ onto $BC$. Prove that the bisector of $\angle BIC$ is perpendicular to the bisector of $\angle AID$.

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: $$a\sqrt b=a+c,\qquad b\sqrt c=b+a,\qquad c\sqrt a=c+b.$$

In an $8\times8$ grid, $n$ disks, numbered 1 to $n$ are stacked, with random order, in a pile in the bottom left corner. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right corner and stack them in the order $1,2,\ldots,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.