# INMO Math Contest Preparation

The Indian National Mathematical Olympiad (INMO) is a math contest in India. This page outlines the contest details and topics covered, providing relevant wikis and quizzes for training and practice.

## Contest Information

## Basic Topics

These are the basic topics covered by the INMO. This is a great place to start learning if you’re new to the INMO!

$\large \textbf{Algebra}$

$\large \textbf{Geometry}$

$\large \textbf{Combinatorics (Counting and Probability)}$

ChapterWikiQuizSet Operations Principle of Inclusion/Exclusion Understanding Data

$\large \textbf{Number Theory}$

## Intermediate Topics

These are the intermediate topics covered by the INMO. This is a great place to hone your skills if you are already comfortable with solving the easiest problems on the INMO.

$\large \textbf{Algebra}$

$\large \textbf{Geometry}$

$\large \textbf{Combinatorics (Counting and Probability)}$

ChapterWikiQuizRules of Sum and Product Discrete Probability Expected Value Combinations

$\large \textbf{Number Theory}$

## Advanced Topics

These are the advanced topics covered by the INMO, which usually appear in the last problems. This is a great place to learn to solve the hardest problems on the INMO if you're shooting for a perfect score!

$\large \textbf{Algebra}$

$\large \textbf{Geometry}$

$\large \textbf{Combinatorics (Counting and Probability)}$

ChapterWikiQuizGeometric Probability Binomial Distribution Conditional Probability

$\large \textbf{Number Theory}$

## Sample Questions

Let $ABC$ be a right triangle with $\angle B = 90^\circ.$ Let $BD$ be the altitude from $B$ onto $AC$. Let $P, Q,$ and $I$ be the incenters of $ABD, CBD,$ and $ABC,$ respectively. Show that the circumcenter of triangle $PIQ$ lies on hypotenuse $AC.$

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

Find all real functions $f$ from $R \rightarrow R$ satisfying the relation $f\big(x^2 + yf(x)\big) = xf(x + y).$

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 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.$)$

Let $ABCD$ be a convex quadrilateral. Let the diagonals $AC$ and $BD$ intersect in $P.$ Let $PE,PF,PG,PH$ be the altitudes from $P$ onto sides $AB, BC, CD, AD,$ respectively. Show that $ABCD$ has an incircle if and only if $\frac{1}{PE} + \frac{1}{PG} = \frac{1}{PF} + \frac{1}{PH}.$

Show that from a set of all 11 integers one can select six numbers $a^2, b^2, c^2, d^2, e^2, f^2$ such that $a^2 + b^2 + c^2 \equiv d^2 + e^2 + f^2 \pmod {12}.$

**Cite as:**INMO Math Contest Preparation.

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