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\noindent {\bf 42nd IMO 2001}
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\noindent {\bf A1}. $ABC$ is acute-angled. $O$ is its circumcenter. $X$ is the foot of the perpendicular from $A$ to $BC. \ \angle C \ge \angle B + 30^o$. Prove that $\angle A + \angle COX < 90^o$.
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\noindent {\bf A2}. $a, b, c$ are positive reals. Prove that ${a \over \sqrt{a^2 + 8bc}} + {b \over \sqrt{b^2 + 8ca}} + {c\over \sqrt{c^2 + 8ab}} \ge 1$.
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\noindent {\bf A3}. Integers are placed in each of the $441$ cells of a $21 \times 21$ array. Each row and each column has at most $6$ different integers in it. Prove that some integer is in at least 3 rows and at least 3 columns.
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\noindent {\bf B1}. Let $n_1, n_2, \ldots , n_m$ be integers where $m$ is odd. Let $x = (x_1, \ldots , x_m)$ denote a permutation of the integers $1, 2, \ldots , m$. Let $f(x) = x_1 n_1 + x_2 n_2 + \cdots + x_m n_m$. Show that for some distinct permutations $a, b$ the difference $f(a) - f(b)$ is a multiple of $m!$.
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\noindent {\bf B2}. $ABC$ is a triangle. $X$ lies on $BC$ and $AX$ bisects $\angle A$. $Y$ lies on $CA$ and $BY$ bisects $\angle B. \ \angle A = 60^o. \ AB + BX = AY + YB$. Find all possible values for $\angle B$.
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\noindent {\bf B3}. $K > L > M > N$ are positive integers such that $KM + LN = (K + L - M + N)(-K + L + M + N)$. Prove that $KL + MN$ is composite.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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