RMO 2011

1. Let $ABC$ be a triangle. Let $D, E, F$ be points respectively on the segments $BC, CA, AB$ such that $AD, BE, CF$ concur at the point $K$. Suppose $\frac{BD}{DC} = \frac{BF}{FA}$ and $∠ADB = ∠AFC$. Prove that $∠ ABE = ∠ CAD$.
2. Let $ (a_1a_2a_3.....a_{2011}) $ be a permutation (that is arrangement) of the numbers $1, 2, 3 . . . , 2011$. Show that there exist two numbers $j, k$. such that $ 1 \le j < k \le 2011 $ and $ |a_{j}-j|=|a_{k}-k| $.
3. A natural number $n$ is chosen strictly between two consecutive perfect square. The smaller of these two squares is obtained by subtracting $k$ from $n$ and the larger one is obtained by adding $l$ to $n$. Prove that $ n-kl $ is a perfect square.
4. Consider a $20$-sided convex polygon K, with vertice $ A_1,A_2,....,A_{20} $ in that order. Find the number of ways in which three sides of  $K$ can be chosen so that every pair among them has at least two sides of  $K$ between them. (For example $ A_1A_2, A_4A_5, A_{11}A_{12} $ is an admissible triple while $ A_1A_2, A_4A_5, A_{19}A_{20} $ is not).
5. Let ABC be a triangle and let $ BB_1,CC_1 $ be respectively the bisectors of ∠ B, ∠ C with $ B_1 $ on AC and $ C_1 $ on AB. Let E, F be the feet of perpendiculars drawn from A onto $ BB_1,CC_1 $ respectively. Suppose $D$ is the point at which the in circle of  $ABC$ touches $AB$. Prove that  $AD = EF$.
6. Find all pairs $(x, y)$ of real numbers such that $ 16^{x^2+y}+16^{x+y^2}=1 $.

Some Useful Links:

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

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