RMO 2009

In this post, there are problems from Regional Mathematics Olympiad, RMO 2009. Try out these problems.

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?

Some Useful Links:

RMO 2002 Problem 2 – Fermat’s Last Theorem as a guessing tool– Video

Our Math Olympiad Program