IMC 2005 — Keystage 3 (IWYMIC) — team

The positive integers $a,b,c$ satisfy $a+b+c=20=ab+bc-ca-b^2$. Determine all possible values of $abc$.

The sum of forty-nine positive integers is $624$. Prove that three of them are equal to one another.

The list $2,3,5,6,7,10,\ldots$ contains in increasing order all positive integers that are neither squares nor cubes. What is its two-thousand-and-fifth number?

$ABCD$ is a convex quadrilateral whose triangle incircles in $BAD$ and $BCD$ are tangent to one another. Prove that $ABCD$ has an incircle.

Find a dissection of a triangle into twenty congruent triangles.

You gamble with the Devil with three dollars initially. Five games are played. Each game you pay an integer number of dollars, from zero to your current funds. If you win, the Devil returns twice your payment; if you lose, he keeps it. Exactly one game is lost, and the Devil chooses which one after seeing your bet. What is the greatest amount you can guarantee after all five games?

A frog starts on a square adjacent to a corner square of a $5\times5$ board. Each hop is to a horizontally or vertically adjacent square, never diagonally. Prove it cannot visit every square exactly once.

If $n^4-4n^3+15n^2-30n+27$ is prime, what are the possible values of the positive integer $n$?

A V-tromino is a $2\times2$ square with one corner square removed. Show that, regardless of which square is omitted from a $7\times7$ board, its remaining part can be covered by sixteen V-trominoes.

Find all positive integers $a_0\gt a_1\gt \cdots\gt a_n\gt 1$ satisfying $$\left(1-\frac1{a_1}\right)+\cdots+\left(1-\frac1{a_n}\right)=2\left(1-\frac1{a_0}\right).$$