# 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 \(B\) on to \(AC\). 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 natural number \(n > 1,\) write the infinite decimal expansion of \(\dfrac{1}{n}\) (for example, we write \(\dfrac{1}{2} = 0.4\overline{9}\) as its infinite decimal expansion, not as \(0.5\)). Determine the length of the non-periodic part of the infinite decimal expansion of \(\dfrac{1}{n}.\)

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

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 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 \(P\) on to sides \(AB, BC, CD,\) and \(AD\) 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 \[a^2 + b^2 + c^2 \equiv d^2 + e^2 + f^2 \quad (mod \quad 12)\]

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

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