RMO 2009

  1. Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.
  2. Show that there is no integer a such that \(a^2-3a-19 \) is divisible by 289.
  3. Show that \(3^{4008}+4^{2009} \) can be written as product of two positive integers each of which is larger than \(2009^{182} \).
  4. Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
  5. A convex polygon \(\Gamma \) is such that the distance between any two vertices of \(\Gamma \) does not exceed 1.
    1. Prove that the distance between any two points on the boundary of \(\Gamma \) does not exceed 1.
    2. If X and Y are two distinct points inside \(\Gamma \), prove that there exists a point Z on the boundary of \(\Gamma \) such that \(XZ+YZ \le 1 \).
  6. In a book with page numbers from 1 to 100, some pages are torn off. The sum of the numbers on the remaining pages is 4949. How many pages are torn off?

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