INMO 2015 (Indian National Math Olympiad)

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 lie 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 \(\textbf{seven} moves? \) (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)

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