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# RMO 2011 1. Let be a triangle. Let be points respectively on the segments such that concur at the point . Suppose and . Prove that .
2. Let be a permutation (that is arrangement) of the numbers . Show that there exist two numbers . such that and .
3. A natural number is chosen strictly between two consecutive perfect square. The smaller of these two squares is obtained by subtracting from and the larger one is obtained by adding to . Prove that is a perfect square.
4. Consider a -sided convex polygon K, with vertice in that order. Find the number of ways in which three sides of can be chosen so that every pair among them has at least two sides of between them. (For example is an admissible triple while is not).
5. Let ABC be a triangle and let be respectively the bisectors of ∠ B, ∠ C with on AC and on AB. Let E, F be the feet of perpendiculars drawn from A onto respectively. Suppose is the point at which the in circle of touches . Prove that .
6. Find all pairs of real numbers such that .

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

1. Let be a triangle. Let be points respectively on the segments such that concur at the point . Suppose and . Prove that .
2. Let be a permutation (that is arrangement) of the numbers . Show that there exist two numbers . such that and .
3. A natural number is chosen strictly between two consecutive perfect square. The smaller of these two squares is obtained by subtracting from and the larger one is obtained by adding to . Prove that is a perfect square.
4. Consider a -sided convex polygon K, with vertice in that order. Find the number of ways in which three sides of can be chosen so that every pair among them has at least two sides of between them. (For example is an admissible triple while is not).
5. Let ABC be a triangle and let be respectively the bisectors of ∠ B, ∠ C with on AC and on AB. Let E, F be the feet of perpendiculars drawn from A onto respectively. Suppose is the point at which the in circle of touches . Prove that .
6. Find all pairs of real numbers such that .

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

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