IMC 2011 — Keystage 3 (IWYMIC) — team

Find all real solutions of $x^2-x+1=(x^2+x+1)(x^2+2x+4)$.

Seventeen $1\times2$ dominoes cover a $5\times8$ board except six cells. Three vacant cells carry white circles in the diagram; the two black-circle cells are occupied. The other three vacant cells are in one vertical column. Which column is it?

Fill the vacant boxes with $1,2,3,4,5,6,7,9,10,11,12,13,14$ once each. Arrows from the box containing zero must point to the box containing one, those from one to two, and so on. The given boxes contain zero, eight and fifteen.

The diagram shows a $5\times8$ board with two marked cells and two example $3\times4$ subboards containing both. How many rectangular subboards of any size contain at least one marked cell?

Three avenues of widths fifteen, fourteen and thirteen metres meet at Red Triangle outside Moscow. Three swinging gates are hinged at junctions A, B, C. As shown, A and B close one avenue while C is pushed aside to allow traffic between the other two. Find the three gate lengths if every pair closes one avenue exactly.

$f(x)$ has degree two thousand ten and $f(k)=-2/k$ for the first two thousand eleven positive integers $k$. Find $f(2012)$.

A cat arranges eighty-one mice in a circle, numbered one through eighty-one clockwise. It counts “one, two, three” clockwise and eats every third mouse, continuing until two remain. If the larger surviving number is forty, at which mouse did counting start?

In triangle $ABC$, $BC=AC$ and $\angle BCA=90^\circ$. Points $D,E$ lie on $AC,AB$ respectively, with $AD=AE$ and $2CD=BE$. Let $P$ be the intersection of $BD$ and the bisector of $\angle CAB$. Find $\angle PCB$ in degrees.

Paint twenty-one of the forty-nine squares of a $7\times7$ board so that no four painted squares form the four corners of any subboard.

Arie, Bert, Caroline know only their own positive integers $a,b,c$, and are told $1/a+1/b+1/c=1$. They are asked: (one) Do you know $a+b+c$? (two) Do you know all three values? Arie answers no to both. Hearing this, Bert answers yes to one and no to two. Caroline hears everything. How does she answer?