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This post contains the six Indian National Maths Olympiad, INMO 2015 problems. Try to solve these problems.

Let ABC be a right-angled triangle with . Let BD is the altitude from B on AC. Let P, Q and Ibe the incenters of triangles ABD, CBD, and ABC respectively. Show that circumcenter of triangle PIQ lies on the hypotenuse AC.

For any natural number n > 1 write the finite decimal expansion of (for example we write as its infinite decimal expansion not 0.5). Determine the length of non-periodic part of the (infinite) decimal expansion of .

Find all real functions such that

There are four basketball 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 (for example , or ).

Let ABCD be a convex quadrilateral. Let diagonals AC and BD intersect at P. Let PE, PF, PG, and PH are altitudes from P on the side AB, BC, CD, and DA respectively. Show that ABCD has a incircle if and only if .

Show that from a set of 11 square integers one can select six numbers such that (mod 12)

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