IMC 2007 — Keystage 3 (IWYMIC) — team

Place the numbers one to nine once each in the nine regions of the Olympic symbol shown, so that the sum inside every circle is fourteen.

Fourteen paper pieces are stacked as shown. Start at $B$, move to adjacent pieces, and finish at $F$, alternating a climb to a higher piece and a descent to a lower piece. Pieces may be revisited; not every piece must be visited. List a valid route.

Label the fourteen intersection points of the seven-pointed star with one to fourteen, each once, so that the four labels on each straight line have the same sum.

A three-digit number has a square ending in the same three digits. Find the sum of all distinct three-digit numbers with this property.

Find all positive integers $m,n$ with $10(m^2+1)=n^2+1$, where $m^2+1$ is prime.

Four teams play each other twice over a six-day week, each team playing one game per day. The shown scoreboard is partly broken into four one-sided printed pieces. Black denotes a win, white a loss. Which team wins the tournament?

Let $A$ be a $3\times3$ array of one to nine, each once. Form $B$ by $b_{ij}=|r_i-c_j|$, where $r_i$ is the sum of row $i$ of $A$, and $c_j$ the sum of column $j$. Can $B$ also contain one to nine each once?

Convex $ABCD$ has perpendicular diagonals $AC,BD$. Through midpoint $M$ of $AB$ draw a line perpendicular to $CD$; through midpoint $N$ of $AD$ draw a line perpendicular to $CB$. Prove their intersection lies on $AC$.

Arrange the integers one to $n$, $n\gt 1$, in a line so every adjacent pair sums to a square. What is the minimum $n$?

The shown $8\times8$ board is partly coloured with five colours: R red, Y yellow, B blue, G green and W white. Complete all sixty-four squares so all squares of each colour are connected edge to edge.