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\noindent {\bf 16th IMO 1974}
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\noindent {\bf A1}. Three players play the following game. There are three cards each with a different positive integer. In each round, the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received $20$, another $10$ and the third $9$ counters. In the last round the player with $10$ received the largest number of counters. Who received the middle number on the first round?
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\noindent {\bf A2}. Prove that there is a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$ iff $\sin A\sin B\le\sin^2{C\over2}$.
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\noindent {\bf A3}. Prove that $\sum_0^n{2n+1\choose2k+1}2^{3k}$ is not divisible by $5$ for any non-negative integer $n$.
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\noindent {\bf B1}. An $8\times8$ chessboard is divided into $p$ disjoint rectangles (along the lines between the squares), so that each rectangle has the same number of white squares as black squares, and each rectangle has a different number of squares. Find the maximum possible value of $p$ and all possible sets of rectangle sizes.
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\noindent {\bf B2}. Determine all possible values of ${a\over a+b+d}+{b\over a+b+c}+{c\over b+c+d}+{d\over a+c+d}$ for positive reals $a,b,c,d$.
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\noindent {\bf B3}. Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. Let $n$ be the number of distinct integer roots to $P(x)=1$ or $-1$. Prove that $n\le d+2$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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