# [Solutions] India Regional Mathematical Olympiad 2006

1. Let $ABC$ be an acute-angled triangle and let $D,E,F$ be the feet of perpendiculars from $A,B,C$ respectively to $BC,CA,AB$. Let the perpendiculars from $F$ to $CB,CA,AD,BE$ meet them in $P,Q,M,N$ respectively. Prove that the points $P,Q,M,N$ are collinear.
2. If $a$ and $b$ are natural numbers such that $a+13b$ is divisible by $11$ and $a+11b$ is divisible by $13$, then find the least possible value of $a+b$.
3. If $a,b,c$ are three positive real numbers, prove that $$\frac {a^{2}+1}{b+c}+\frac {b^{2}+1}{c+a}+\frac {c^{2}+1}{a+b}\ge 3$$
4. A $6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
5. Let $ABCD$ be a quadrilateral in which $AB$ is parallel to $CD$ and perpendicular to $AD$, $AB = 3CD$, and the area of the quadrilateral is $4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.
6. Prove that there are infinitely many positive integers $n$ such that $n(n+1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $a^{2}+b^{2}$ and $b^{2}+a^{2}$ are considered as the same representation)
7. Let $X$ be the set of all positive integers greater than or equal to $8$ and let $f: X\rightarrow X$ be a function such that $f(x+y)=f(xy)$ for all $x\ge 4, y\ge 4 .$ if $f(8)=9$, determine $f(9) .$
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