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\noindent {\bf 21st IMO 1979}
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\noindent {\bf A1}. Let $m$ and $n$ be positive integers such that $${m\over n}=1-{1\over2}+{1\over3}-{1\over4}+\ldots-{1\over1318}+{1\over1319}.$$ Prove that $m$ is divisible by $1979$.
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\noindent {\bf A2}. A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as the top and bottom faces is given. Each side of the two pentagons and each of the $25$ segments $A_iB_j$ is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Prove that all $10$ sides of the top and bottom faces have the same color.
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\noindent {\bf A3}. Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each traveling along is own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P$.
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\noindent {\bf B1}. Given a plane $k$, a point $P$ in the plane and a point $Q$ not in the plane, find all points $R$ in $k$ such that the ratio $QP+PR\over QR$ is a maximum.
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\noindent {\bf B2}. Find all real numbers $a$ for which there exist non-negative real numbes $x_1,x_2,x_3,x_4,x_5$ satisfying:

$x_1+2x_2+3x_3+4x_4+5x_5=a,$

$x_1+2^3x_2+3^3x_3+4^3x_4+5^3x_5=a^2,$

$x_1+2^5x_2+3^5x_3+4^5x_4+5^5x_5=a^3.$
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\noindent {\bf B3}. Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A$. From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: $a_{2n-1}=0,a_{2n}={(2+\sqrt2)^{n-1}\over\sqrt2}-{(2-\sqrt2)^{n-1}\over\sqrt2}$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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