- Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.
- Show that there is no integer a such that is divisible by 289.
- Show that can be written as product of two positive integers each of which is larger than .
- Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
- A convex polygon is such that the distance between any two vertices of does not exceed 1.
- Prove that the distance between any two points on the boundary of does not exceed 1.
- If X and Y are two distinct points inside , prove that there exists a point Z on the boundary of such that .
- 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?