30th Indian National Mathematical Olympiad 2015
Let ABC be a right angled triangle with angle B = 90. Let BD be the altitude from B on to AC. Let P, Q and I be incenters of triangles ABD, CBD and ABC respectively. Show that the circumcenter of triangle PIQ lies on hypotenuse AC.
For any natural number n > 1, write the infinite decimal expansion of 1/n (for example, we write 1/2 = 0.4999... as its infinite decimal expansion, not 0.5). Determine the length of non-periodic part of the infinite decimal expansion of 1/n.
Find all real functions f from R to R satisfying the relation f(x^2 + yf(x)) = xf(x + y).
There are four basket-ball players A, B, C, D. Initially the ball is with A. The ball is always passed from one person to a different person. In how many ways can the ball come back to A after seven passes? (For example A -> C -> B -> D -> A -> B -> C -> A and A -> D -> A -> D -> C -> A -> B -> A are two ways in which the ball can come before to A after seven passes.)
Let ABCD be a convex quadrilateral. Let the diagonals AC and BD intersect in P. Let PE, PF, PG, PH be the altitudes from P on to sides AB, BC, CD and AD respectively. Show that ABCD has incircle iff 1/PE + 1/PF = 1/PF + 1/PH.
Show that from a set of all 11 integers one can select six numbers a^2, b^2, c^2, d^2, e^2, f^2 such that a^2 + b^2 + c^2 == d^2 + e^2 + f^2 (mod 12).