INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

February 5, 2021

Indian National Math Olympiad, INMO 2015 Problems


This post contains problems from Indian National Mathematics Olympiad, INMO 2015. Try them and share your solution in the comments.

INMO 2015, Problem 1

Let $A B C$ be a right-angled triangle with $\angle B=90^{\circ} .$ Let $B D$ be the altitude from $B$ on to $A C .$ Let $P, Q$ and $I$ be the incentres of triangles $A B D, C B D$ and $A B C$ respectively. Show that the circumcentre of of the triangle $P I Q$ lies on the hypotenuse $A C$.

INMO 2015, Problem 2

For any natural number $n>1$, write the infinite decimal expansion of $1 / n$ (for example, we write $1 / 2=0.4 \overline{9}$ as its infinite decimal expansion, not 0.5 ). Determine the length of the non-periodic part of the (infinite) decimal expansion of $1 / n$.

INMO 2015, Problem 3

Find all real functions $f$ from $\mathbb{R} \rightarrow \mathbb{R}$ satisfying the relation
$$f\left(x^{2}+y f(x)\right)=x f(x+y)$$

INMO 2015, Problem 4

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 $A \rightarrow C \rightarrow B \rightarrow D \rightarrow A \rightarrow B \rightarrow C \rightarrow A$ and $A \rightarrow D \rightarrow A \rightarrow D \rightarrow C \rightarrow A \rightarrow B \rightarrow A$ are two ways in which the ball can come back to $A$ after seven passes).

INMO 2015, Problem 5

Let $A B C D$ be a convex quadrilateral. Let the diagonals $A C$ and $B D$ intersect in $P$. Let $P E, P F, P G$ and $P H$ be the altitudes from $P$ on to the sides $A B, B C, C D$ and $D A$ respectively. Show that $A B C D$ has an incircle if and only if
$$\frac{1}{P E}+\frac{1}{P G}=\frac{1}{P F}+\frac{1}{P H}$$

INMO 2015, Problem 6

From a set of 11 square integers, show that one can choose 6 numbers $a^{2}, b^{2}, c^{2}, d^{2}, e^{2}, f^{2}$ such that
$$a^{2}+b^{2}+c^{2} \equiv d^{2}+e^{2}+f^{2} \quad(\bmod 12)$$

Some Useful Links

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter