 1. In this post, there are problems from Regional Mathematics Olympiad, RMO 2009. Try out these problems.
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3. Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.
Discussion
4. Show that there is no integer a such that $a^2-3a-19$ is divisible by 289.
5. Show that $3^{4008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$.
6. Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
7. 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 $XZ+YZ \le 1$.
8. 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?