INMO Practice Test-I
Time: 4 Hours [100 marks]
The real numbers a, b, c, d satisfy simultaneously the equations
Prove that 
Consider a triangle ABC and let M be the midpoint of the side BC.
Suppose and . Find the measure of 
There are two piles of coins, each containing 2010 pieces. Two players A and B play a game taking turns (A plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
Find all positive integers n such that is a product of two or more consecutive positive integers. 
Consider a triangle with . Let F be the foot of the altitude from C. Circle touches the line segment at point , the altitude at point Q and the circumcircle of ABC at point R. Prove that points are collinear and . 
The real positive numbers satisfy the relations Prove that
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