IMC 2006 — Keystage 3 (IWYMIC) — team

A teacher wants the largest possible circle inside a triangle with sides $2,2,2x$, for real $x\gt 0$. Alex proposes $x=1$; Brian proposes $x=\sqrt2$; Colin says both are wrong. Who is right?

A triangle can be cut into two isosceles triangles. One original angle is $36^\circ$. Determine all possible values, in degrees, of the original triangle’s largest angle.

Five tetrominoes each consist of four unit squares joined edge to edge. Combine the L-shaped piece with each of the other four shown pieces to form a shape with a reflection axis.

Fifteen dominoes, labelled $11,12,13,14,15,22,23,24,25,33,34,35,44,45,55$, fill the shown $5\times6$ rectangle. Each domino consists of two numbered squares sharing an edge. Their individual boundaries have been erased. Reconstruct them.

A lucky number is a positive integer equal to nineteen times the sum of its decimal digits. Determine all lucky numbers.

Alice and Betty alternately enter zero or one in any blank square of an $n\times n$ board, Alice first. Once full, Betty wins if every row has even sum; otherwise Alice wins. (a) Who has a winning strategy for $n=2006$? (b) Answer for every positive integer $n$.

Prove that $1596^n+1000^n-270^n-320^n$ is divisible by $2006$ for every positive odd integer $n$.

Delete from the positive integers all multiples of four and all numbers one more than a multiple of four. Let $S_n$ be the sum of the first $n$ remaining terms. Compute $$\lfloor\sqrt{S_1}\rfloor+\lfloor\sqrt{S_2}\rfloor+\cdots+\lfloor\sqrt{S_{2006}}\rfloor,$$ where the floor is the greatest integer not exceeding its argument.

$ABC,PQR$ are equilateral triangles, each of area one square cm. The centre $M$ of $PQR$ lies on the perimeter of $ABC$. Find the minimum intersection area in square cm.

For positive integer $m$, some positive integer $n$ makes $mn$ a square and $m-n$ prime. Find all such $m$ with $1000\le m\lt 2006$.