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# RMO 2010 | Regional Mathematics Olympiad Problems In this post, there are problems from Regional Mathematics Olympiad, RMO 2010. Try out these problems.

1. Let be a convex hexagon in which the diagonals are concurrent at . Suppose the area of triangle is the geometric mean of those of and ; and the area of the triangle is the geometric mean of those of and . Prove that the area of triangle is the geometric mean of those of and .
2. Let , , be three quadratic polynomials where are non-zero real numbers. Suppose there exits a real number such that . Prove that .
3. Find the number of -digit numbers (in base ) having non-zero digits and which are divisible by but not by .
4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
5. Let ABC be a triangle in which Let and be the bisectors of the angles and with on and on . Let be the reflection of in the line . Prove that lies on .
6. For each integer 1 ≤ n, define where [x] denotes the largest integer not exceeding x, for any real numbers x. Find the number of all n in the set for which .

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

In this post, there are problems from Regional Mathematics Olympiad, RMO 2010. Try out these problems.

1. Let be a convex hexagon in which the diagonals are concurrent at . Suppose the area of triangle is the geometric mean of those of and ; and the area of the triangle is the geometric mean of those of and . Prove that the area of triangle is the geometric mean of those of and .
2. Let , , be three quadratic polynomials where are non-zero real numbers. Suppose there exits a real number such that . Prove that .
3. Find the number of -digit numbers (in base ) having non-zero digits and which are divisible by but not by .
4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
5. Let ABC be a triangle in which Let and be the bisectors of the angles and with on and on . Let be the reflection of in the line . Prove that lies on .
6. For each integer 1 ≤ n, define where [x] denotes the largest integer not exceeding x, for any real numbers x. Find the number of all n in the set for which .

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

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