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# 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

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