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I.S.I. and C.M.I. Entrance

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 \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)

By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

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