IMC 2009 — Keystage 3 (IWYMIC) — team

Cards one to fifteen form a deck. Repeatedly put the top card on the table pile, then move the next top card to the bottom of the remaining deck. Continue until all cards are on the table. That pile reads one to fifteen from top to bottom. What was the fourth card from the bottom of the original deck?

Find the smallest positive integer having ten divisors ending in all ten different decimal digits. It may have more than ten divisors.

Complete the shown $6\times6$ table so each digit one through six appears in every row, column and both main diagonals.

The town map shows three parking areas P and seven shaded squares. One-way arrows apply up to the first side street. Find a route starting at one parking area, visiting all seven squares and ending at a different parking area, without visiting any point twice, including intersections.

Draw a closed horizontal-and-vertical path through every cell of the board exactly once without crossing itself. Between consecutive circles of the same colour, it must use one straight segment; between different colours, exactly two straight segments meeting at a right angle.

For positive integers $n$, let $a_n=\frac{2^n}{2^{2n+1}-2^{n+1}-2^n+1}$. Prove $a_1+\cdots+a_{2009}\lt 1$.

Find every partition of the positive integers into hot and cold numbers such that a hot plus a cold number is hot, and their product is cold.

A regular octagon of side one cm is cut as shown into a square and four congruent pentagons, then rearranged into a square. Find one pentagon’s perimeter in cm.

Each card game has four players. The number of games equals the number of contestants, and every pair plays together in at least one game. What is the maximum number of contestants?

Which of $2008,2009,2010$ can be written $x^3+y^3+z^3-3xyz$ with positive integers $x,y,z$?