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.

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

$$\large \textbf{Algebra}$$

Chapter	Wiki	Quiz
Radical Expressions and Equations
Linear Equations
Systems of Linear Equations
Remainder Factor Theorem
Exponential Inequalities
Functions
Parametric Equations
De Moivre's Theorem
Arithmetic Progressions
Solving Equations

$$\large \textbf{Geometry}$$

Chapter	Wiki	Quiz
Trigonometric Functions
Tangent and Secant Lines
Inscribed and Circumscribed Figures
Equation of a Line
Triangle Centers

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

Chapter	Wiki	Quiz
Set Operations
Principle of Inclusion/Exclusion
Understanding Data

$$\large \textbf{Number Theory}$$

Chapter	Wiki	Quiz
Divisibility
Prime Numbers
Prime Factorization and Divisors
Number Bases
Rational Numbers

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.

$$\large \textbf{Algebra}$$

Chapter	Wiki	Quiz
Absolute Value
Completing the Square
Quadratic Discriminant
Linear Inequalities
Polynomial inequalities
Absolute Value Inequalities
Rational Functions
Geometric Progressions
System of Equations
Quadratic Equations
Function Graphs

$$\large \textbf{Geometry}$$

Chapter	Wiki	Quiz
Solving Triangles
Conic Sections: Parabola
Conic Sections: Ellipse
3D Coordinate Geometry
Graphs of Trig Functions
Fundamental Trig Identities

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

Chapter	Wiki	Quiz
Rules of Sum and Product
Discrete Probability
Expected Value
Combinations

$$\large \textbf{Number Theory}$$

Chapter	Wiki	Quiz
Linear Recurrence Relations
Factorials
Linear Diophantine Equations
Quadratic Diophantine Equations
Modular Arithmetic

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!

$$\large \textbf{Algebra}$$

Chapter	Wiki	Quiz
Binomial Theorem
Rational Root Theorem
Functional Equations
AM-GM Inequality
Exponential Functions
Logarithmic Functions
Complex Numbers
Polar Coordinates
Vieta's Formula
Roots of Unity
Sequences and Series

$$\large \textbf{Geometry}$$

Chapter	Wiki	Quiz
Conic Sections: Hyperbola
Trigonometric Equations
Sum and Difference Trig Formulas
Dot Product of Vectors
Cross Product of Vectors

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

Chapter	Wiki	Quiz
Geometric Probability
Binomial Distribution
Conditional Probability

$$\large \textbf{Number Theory}$$

Chapter	Wiki	Quiz
Quadratic Residues
Arithmetic Functions
Modular Arithmetic Applications
Euler's Theorem
General Diophantine Equations

1. 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 $AC.$

2. 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 $\frac{1}{n}.$

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

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 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.$)$

5. 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 $$\frac{1}{PE} + \frac{1}{PG} = \frac{1}{PF} + \frac{1}{PH}.$$

6. 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 $$a^2 + b^2 + c^2 \equiv d^2 + e^2 + f^2 \pmod {12}.$$