21st inmo 06
- 1. INMO–2006 1. In a nonequilateral triangle ABC, the sides a, b, c form an arithmetic progression. Let I and O denote the incentre and circumcentre of the triangle respectively. (i) Prove that IO is perpendicular to BI. (ii) Suppose BI extended meets AC in K, and D, E are the mid- points of BC, BA respectively. Prove that I is the circumcentre of triangle DKE. 2. Prove that for every positive integer n there exists a unique ordered pair (a, b) of positive integers such that n = 1 2 (a + b − 1)(a + b − 2) + a. 3. Let X denote the set of all triples (a, b, c) of integers. Define a function f : X → X by f(a, b, c) = (a + b + c, ab + bc + ca, abc). Find all triples (a, b, c) in X such that f(f(a, b, c)) = (a, b, c). 4. Some 46 squares are randomly chosen from a 9 × 9 chess board and are coloured red. Show that there exists a 2 × 2 block of 4 squares of which at least three are coloured red. 5. In a cyclic quadrilateral ABCD, AB = a, BC = b, CD = c, ∠ABC = 120◦, and ∠ABD = 30◦. Prove that (i) c ≥ a + b; (ii) | √ c + a − √ c + b| = √ c − a − b. 6. (a) Prove that if n is a positive integer such that n ≥ 40112, then there exists an integer l such that n < l2 < (1 + 1 2005 )n. (b) Find the smallest positive integer M for which whenever an in- teger n is such that n ≥ M, there exists an integer l, such that n < l2 < (1 + 1 2005 )n.