# [Solutions] India Regional Mathematical Olympiad 2001

1. Let $BE$ and $CF$ be the altitudes of an acute triangle $ABC$ with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively. such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$. Prove that $FM$ is parallel to $EN$.
2. Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square.
3. Find the number of positive integers $x$ such that $\left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] .$
4. Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.
5. In a triangle $ABC$, $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD =AB$. prove that $\angle A = 72^{\circ}$.
6. If $x,y,z$ are sides of a triangle, prove that $| x^2(y-z) + y^2(z-x) + z^2(x-y) | < xyz.$
7. Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ positive odd integers by a multiple of $2001$.
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