IMC 2010 — Keystage 3 (IWYMIC) — team

Solve for real numbers $w,x,y,z$: $$\begin{cases}w+8x+3y+5z=20,\\4w+7x+2y+3z=-20,\\6w+3x+8y+7z=20,\\7w+2x+7y+3z=-20.\end{cases}$$

In convex quadrilateral $ABCD$, $AB$ is the shortest side and $CD$ the longest. Prove $\angle A\gt \angle C$ and $\angle B\gt \angle D$.

Integers $m\ge n$ satisfy $m^3+n^3+1=4mn$. Find the maximum of $m-n$.

Sixty-four dots form an $8\times8$ array with adjacent same-row or same-column dots one centimetre apart. Count all rectangles of area twelve square centimetres whose vertices are among the dots.

Find the largest positive integer $n$ for which exactly one positive integer $k$ satisfies $\frac8{15}\lt \frac{n}{n+k}\lt \frac7{13}$.

Each row and column of a $9\times9$ table contains at most four distinct numbers. What is the maximum total number of distinct numbers in the table?

In convex quadrilateral $ABCD$, $\angle ADB=16^\circ$, $\angle BDC=48^\circ$, $\angle ACD=58^\circ$, $\angle BCA=30^\circ$. Find $\angle ABD$.

Determine all ordered triples $(x,y,z)$ of positive real numbers for which each of $x+1/y$, $y+1/z$, $z+1/x$ is an integer.

Put the integers one through fifteen into the fifteen circles in the left diagram, using each once. Each circle’s value plus all values in touching circles must equal the corresponding number in the right diagram. Each row except the first must also have the indicated row sum.

The letters K, O, R, E, A, I, M, C form eight rows, with one K in the first, two Os in the second, and so on to eight Cs in the last. Starting at K, spell KOREAIMC by moving to the next row, directly below or to either neighbour, as shown by the bold path. One of the thirty-six letters is forbidden, reducing the number of paths to five hundred sixteen. Which letter is forbidden?