Download [PDF] 2014 SMO Senior Section Round 2 Questions Singapore Mathematical Olympiad

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SMO Round 2 2014 Senior 28 June 2014 1. In the triangle ABC, the excircle opposite to the vertex Awith centre at Itouches the side BCatD. (The circle also touches the sides AB; ACex- tended.) Let Mbe the midpoint of BCand Nthe midpoint of AD. Prove that I ; M ; N are collinear. 2. Find, with justi cation, all positive real numbers a; b; csatisfying the system of equations: ap b = a+ c; b p c = b+ a; c p a = c+ b: 3. Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk tan the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red. Suppose the bottom level has 2048 disks of which 2014 are red, What is the colour of the disk at the top? 4. For each positive integer n, let x n = p 1 + : : : +p n where p 1; : : : ; p nare the rst few nprimes. Prove that for each positive integer n, there is an integer k n such that x n < k n2 < x n+1 . 5. Alice and Bob play a number game. Starting with a positive integer n, they take turns changing the numvber with Alice going rst. Each player may change the current number kto either k 1 or dk= 2e. The person who changes 1 to 0 wins. Determine all nsuch that Alice has a winnin strategy. (For each real number x, its ceiling dx e is the smallest intger x. 1