Bìa sách [Solutions] India Regional Mathematical Olympiad 2006
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[Solutions] India Regional Mathematical Olympiad 2006

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Giới thiệu
  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 imes 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 ightarrow X$ be a function such that $ f(x+y)=f(xy)$ for all $ xge 4, yge 4 .$ if $ f(8)=9$, determine $ f(9) .$