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RMO 2009

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 $$a^2-3a-19$$ 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.
1. Prove that the distance between any two points on the boundary of $$\Gamma$$ does not exceed 1.
2. 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?
October 20, 2013