Question bank xIMC 2012

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 $\angle CFE$.

A positive real number is given. Each move adds three, subtracts three, multiplies by three or divides by three. Determine every starting number which can return to itself after exactly three moves.

Eight people have average age fifteen; every age is prime. Nineteen occurs more often than any other age. In sorted order the two middle ages average eleven. Find the oldest person’s maximum age.

The digits one through nine occupy the nine hexagons on the sides of a triangle, initially summing to nineteen on each side. Rearrange them so the three side sums remain equal. Find the minimum and maximum common sum.

Two thousand twelve equally spaced collinear points are painted orange or green. If three points $A,B,C$ satisfy $AB=BC$, all three must have the same colour. Count all colourings.

The three splits $2|012$, $20|12$, $201|2$ of the four-digit number two thousand twelve give product sum $2\cdot012+20\cdot12+201\cdot2=666$. Find all other four-digit numbers with this property.