20th inmo 05
- 1. INMO–2005 February 6, 2005 1. Let M be the midpoint of side BC of a triangle ABC. Let the median AM intersect the incircle of ABC at K and L, K being nearer to A than L. If AK = KL = LM, prove that the sides of triangle ABC are in the ratio 5 : 10 : 13 in some order. 2. Let α and β be positive integers such that 43 197 < α β < 17 77 . Find the minimum possible value of β. 3. Let p, q, r be positive real numbers, not all equal, such that some two of the equations px2 + 2qx + r = 0, qx2 + 2rx + p = 0, rx2 + 2px + q = 0, have a common root, say α. Prove that (a) α is real and negative; and (b) the remaining third equation has non-real roots. 4. All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g.,877550) are written as a sequence in increasing order. Find the 2005-th number in this sequence. 5. Let x1 be a given positive integer. A sequence(xn)∞ n=1 = (x1, x2, x3, · · ·) of positive integers is such that xn, for n ≥ 2, is obtained from xn−1 by adding some nonzero digit of xn−1. Prove that (a) the sequence has an even number; (b) the sequence has infinitely many even numbers. 6. Find all functions f : R → R such that f(x2 + yf(z)) = xf(x) + zf(y), for all x, y, z in R. (Here R denotes the set of all real numbers.) 1