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\noindent {\bf 36th IMO 1995}
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\noindent {\bf A1}. Let $A, B, C, D$ be four distinct points on a line, in that order. The circles with diameter $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM, DN, XY$ are concurrent.
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\noindent {\bf A2}. Let $a, b, c$ be positive real numbers with $abc = 1$. Prove that: $${1 \over a^3(b+c)} + {1 \over b^3(c+a)} + {1 \over c^3(a+b)} \ge {3 \over 2}.$$
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\noindent {\bf A3}. Determine all integers $n>3$ for which there exist $n$ points $A_1, \ldots , A_n$ in the plane, no three collinear, and real numbers $r_1, \ldots , r_n$ such that for any distinct $i, j, k$, the area of the triangle $A_iA_jA_k$ is $r_i+r_j+r_k$. 
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\noindent {\bf B1}. Find the maximum value of $x_0$ for which there exists a sequence $x_0, x_1, \ldots , x_{1995}$ of positive reals with $x_0 = x_{1995}$ such that for $i = 1, \ldots , 1995$: $$x_{i-1}+{2 \over x_{i-1}} = 2x_i+{1 \over x_i}.$$
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\noindent {\bf B2}. Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD = \angle EFA = 60^o$. Suppose that $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB = \angle DHE = 120^o$. Prove that $AG+GB+GH+DH+HE \ge CF$.
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\noindent {\bf B3}. Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1, 2, \ldots , 2p\}$ are there, the sum of whose elements is divisible by $p$?
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 23 August 2003

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