INMO 2015 Problems | Indian National Maths Olympiad

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

1. Let ABC be a right-angled triangle with $ \angle{B}=90^{\circ} $. 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.
2. For any natural number n > 1 write the finite decimal expansion of $ \frac{1}{n} $ (for example we write $ \frac{1}{2}=0.4\overline{9} $ as its infinite decimal expansion not 0.5). Determine the length of non-periodic part of the (infinite) decimal expansion of $ \frac{1}{n} $.
3. Find all real functions $ f: \mathbb{R} to \mathbb{R} $ such that $ f(x^2+yf(x))=xf(x+y) $
4. 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 seven moves? $ (for example $ A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A $ , or $ A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A $).
5. 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 $ \frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH} $.
6. Show that from a set of 11 square 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 \equiv d^2+e^2+f^2 $ (mod 12)