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 a2 − 3a − 19 is divisible by 289.
3. Show that 32008 + 42009 can be wriiten as product of two positive integers each of which is larger than 2009182.
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 Γ is such that the distance between any two vertices of Γ does not exceed 1.
(i) Prove that the distance between any two points on the boundary of Γ does not exceed 1.
(ii) If X and Y are two distinct points inside Γ, prove that there exists a point Z on the boundary of Γ such that XZ + Y Z ≤ 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?