IMC 2012 — Keystage 3 (IWYMIC) — individual

Find the maximum difference of two positive integers whose sum is two thousand thirty-four and whose product is a multiple of that sum.

A semicircle sits on top of a square and is tangent to the two sloping sides of an equilateral triangle, as shown; the triangle and square bases lie on the same line. If each triangle side is twelve centimetres, find the semicircle radius.

Find the second smallest four-digit multiple $\overline{abcd}$ of eleven with $b+c=a$ and two-digit number $\overline{bc}$ an integer square.

Equilateral triangle $ABC$ has area $8+4\sqrt3$ square centimetres. $M$ is the midpoint of $BC$. The bisector of $\angle MAB$ meets $BM$ at $N$. Find the area of $ABN$.

A $2\times6$ hole in a wall is filled with unit tiles coloured red, white or blue. No two tiles of the same colour may share a side. Count all fillings.

How many positive divisors of $1^9\cdot2^8\cdot3^7\cdot4^6\cdot5^5\cdot6^4\cdot7^3\cdot8^2\cdot9$ are perfect squares?

How many positive integers at most twenty million one hundred twelve thousand twelve use only the digits zero, one or two?

Points $A,B,C,D$ lie on a circle. $E$ is on the extension of $BA$ and $AD$ bisects $\angle CAE$. $F$ lies on $AC$, with $DF\perp AC$. If $BA=AF=2$ cm, find $AC$.

Each digit of the four-digit number $\overline{abcd}$ is one, two, three or four. In how many of the two hundred fifty-six possibilities is $ad-bc$ even?

Twenty-four equally spaced points lie on a circle. How many equilateral triangles have at least two vertices among those points? The third vertex need not lie on the circle.

Sector $OAB$ is one-sixth of a circle. An inner circle touches $OA,OB$ and arc $AB$ as shown. What fraction of the sector area does the inner circle occupy?

Three identical darts hit different cells of an $8\times8$ chessboard. Every pair of hit cells shares at least one vertex. How many different sets of cells can be hit?

Find the integral part of $$\sqrt{2012\sqrt{2013\sqrt{\cdots\sqrt{(2012^2-2)\sqrt{(2012^2-1)\sqrt{2012^2}}}}}}.$$ The integer coefficients increase by one up to the innermost value $2012^2$.

Positive integers $m,n$ satisfy $n^2\lt 8m\lt n^2+60(\sqrt{n+1}-\sqrt n)$. Find the maximum possible $n$.

In triangle $ABC$, $\angle A=90^\circ$, $\angle B=20^\circ$. Points $E,F$ lie on $AC,AB$, respectively, with $\angle ABE=10^\circ$, $\angle ACF=30^\circ$. Find $\angle CFE$.