1. Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.
    Discussion
  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.
    • Prove that the distance between any two points on the boundary of \Gamma does not exceed 1.
    • 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?