RMO 2015 Mumbai Region | Problem and Solutions

This post contains Regional Mathematics Olympiad, RMO 2015 Mumbai Region problems, and solutions

1. Let ABCD be a convex quadrilateral with AB = a, BC = b, CD = c and DA = d. Suppose $ a^2 + b^2 + c^2 + d^2 = ab + bc + cd + da $ and the area of ABCD is 60 square units. If the length of one of the diagonals is 30 unit, determine the length of the other diagonal.
   SOLUTION: here
2. Determine the number of 3 digit numbers in base 10 having at least one 5 and at most one 3.
   SOLUTION: here
3. Let P(x) be a polynomial whose coefficients are positive integers. If P(n) divides P(P(n) -2015) for every natural number n, prove that P(-2015) = 0.
   SOLUTION: here
4. Find all three digit natural numbers of the form $ (abc)_{10} $ and $ s=2$ such that $ (abc)_{10} , (bca)_{10} ,(cab)_{10}$ and $ s=2$ are in geometric progression. (Here $ (abc)_{10} $ and $ s=2$ is representation in base 10).
5. Let ABC be a right angled triangle with $ \angle B = 90^0 $ and $ s=2 $ and let BD be the altitude from B on to AC. Draw $ DE \perp AB $ and $ s=2$ and $ DF \perp BC $ and $ s=2 $. Let P, Q, R and S be respectively the incenters of triangle DFC, DBF, DEB and DAE. Suppose S, R, Q are collinear. Prove that P, Q, R, D lie on a circle.
   SOLUTION: here
6. Let S = {1, 2, ..., n} and let T be the set of all ordered triples of subsets, say $ (A_1, A_2, A_3) $ and $ s=2 $ , such that $ A_1 \cup A_2 \cup A_3 = S $ and $ s=2 $. Determine, in terms of n, $ \sum_{(A_1, A_2, A_3) \in T} |A_1 \cap A_2 \cap A_3 | $ and $ s=2 $, where |X| denotes the number of elements in the set X. (For example, if S={1, 2, 3} and $ A_1={1,2}, A_2={2, 3}, A_3 = {3} $ and $ s=2 $ then one of the elements of T is ({1,2}, {2,3}. {3}).)
7. Let x, y, z be real numbers such that $ x^2 + y^2 + z^2 - 2xyz = 1 $ and $ s=2 $. Prove that $ (1+x)(1+y)(1+z) \le 4 + 4xyz $ and $ s=2 $
   SOLUTION: here
8. The length of each side of a convex quadrilateral ABCD is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.

(Thanks to Eeshan Banerjee for supplying the question paper).