IMC 2002 — Keystage 3 (IWYMIC) — team

Let $m,n,p$ be real numbers. If $a=x^{m+n}y^p$, $b=x^{n+p}y^m$, $c=x^{p+m}y^n$, find $a^{m-n}b^{n-p}c^{p-m}$.

Let $f(x)=\frac{bx+1}{2x+a}$, where $a,b$ are constants with $ab\ne2$. (a) If $f(x)f(1/x)=k$ for all admissible $x$, find $k$. (b) Prove that $f(f(x))$ is not a constant function.

Prove or disprove that a rectangle can be formed using an odd number of copies of the figure in the diagram.

Find all nonzero integers $x,y$ with $x\le y$ and $\frac1x+\frac1y=\frac1{14}$.

Four brothers divide $137$ gold coins, no two receiving the same number. Each receives an integral multiple of the number received by his next younger brother. Find all possible shares.

In triangle $ABC$, $AB=BC$. A line through $B$ meets $AC$ at $D$ so that the inradius of $ABD$ equals the exradius of $CBD$ opposite $B$. Prove that this common radius is one quarter of the altitude from $C$ to $AB$.

Two circles of radii $a$ cm and $b$ cm touch externally. A third circle of radius $c$ cm touches both and one common tangent. Prove $\frac1{\sqrt c}=\frac1{\sqrt a}+\frac1{\sqrt b}$.

Robbie is locked in a solar panel and must escape through the shaded central square. He controls the other dummy robots. A robot can move only along a row or column directly toward another robot, stopping at the empty square immediately before that robot. In each of the eight scenarios, four moves are allowed; a continuous sequence of motions by the same robot counts as one move. Robbie is denoted by $R$. Give a solution for each scenario.