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{\Large\bf XV Asian Pacific Mathematics Olympiad \\
March 2003}
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Time allowed: 4 hours
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No calculators are to be used
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Each question is worth 7 points
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{\large\bf Problem 1}.
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Let $a,b,c,d,e,f$ be real numbers such that the polynomial
$$p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f$$
factorises into eight linear factors $x-x_i$, with $x_i>0$ for
$i=1,2,\ldots,8$. Determine all possible values of $f$.
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{\large\bf Problem 2}.
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Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a
plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$
units apart. The square $ABCD$ is placed on the plane so that sides $AB$
and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$
and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters
of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively.
Prove that no matter how the square was placed, $m_1+m_2$ remains constant.
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{\large\bf Problem 3}.
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Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number
which is strictly less than $k$. You may assume that $p_k\ge 3k/4$. Let $n$
be a composite integer. Prove:
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(a) if $n=2p_k$, then $n$ does not divide $(n-k)!\;$;
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(b) if $n>2p_k$, then $n$ divides $(n-k)!\;$.
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{\large\bf Problem 4}.
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Let $a,b,c$ be the sides of a triangle, with $a+b+c=1$, and let $n\ge 2$ be
an integer. Show that
$$\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+
{\sqrt[n]{2}\over 2}\ .$$
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{\large\bf Problem 5}.
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Given two positive integers $m$ and $n$, find the smallest positive integer
$k$ such that among any $k$ people, either there are $2m$ of them who form
$m$ pairs of mutually acquainted people or there are $2n$ of them forming
$n$ pairs of mutually unacquainted people.
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