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\noindent {\bf 37th IMO 1996}
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\noindent {\bf A1}. We are given a positive integer $r$ and a rectangular board divided into $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$. The task is to find a sequence of moves leading between two adjacent corners of the board which lie along the long side.

\noindent (a) Show that the task cannot be done if $r$ is divisible by $2�$ or $3$.

\noindent (b) Prove that the task is possible for $r = 73$.

\noindent (c) Can the task be done for $r = 97$?
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\noindent {\bf A2}. Let $P$ be a point inside the triangle $ABC$ such that $\angle APB - \angle ACB = \angle APC - \angle ABC$. Let $D, E$ be the incenters of triangles $APB, APC$ respectively. Show that $AP, BD, CE$ meet at a point.
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\noindent {\bf A3}. Let $S$ be the set of non-negative integers. Find all functions $f:S\to S$ such that $f(m+f(n)) = f(f(m))+f(n)$ for all $m,n$.
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\noindent {\bf B1}. The positive integers $a,b$ are such that $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken by the smaller of these two squares?
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\noindent {\bf B2}. Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE, BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A, R_C, R_E$ denote the circumradii of triangles $FAB, BCD, DEF$ respectively, and let $p$ denote the perimeter of the hexagon. Prove that: $$R_A+R_C+R_E\ge p/2$$.
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\noindent {\bf B3}. Let $p,q,n$ be three positive integers with $p+q < n$. Let $x_0, x_1, \ldots , x_n$ be integers such that $x_0 = x_n = 0$, and for each $1\le i \le n, x_i - x_{i-1} = p$ or $-q$. Show that there exist indices $i<j$ with $(i,j)$ not $(0,n)$ such that $x_i=x_j$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 21 August 2003

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