RMO Math Contest Preparation

The Regional Math Olympiad (RMO) is a math contest in India. This page outlines the contest details and topics covered, providing relevant wikis and quizzes for training and practice.

The Pre-RMO is a state level test that determines who can go on to take the RMO. Mostly the cut-off is 50%. But this year in West Bengal, it was 30 out of 80. So it depends on the difficulty level of the questions. The Pre-RMO exam consists of 30 questions, each answer being an integer from 1 to 99. The cut-off for Pre-RMO was 11/30 in Chandigarh.

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

$$\large \textbf{Algebra}$$

Chapter	Wiki	Quiz
Radical Expressions and Equations
Linear Equations
Systems of Linear Equations
Remainder-Factor Theorem
Exponential Inequalities
Functions
Arithmetic Progressions
Solving Equations
Absolute Value
Completing the Square
Polynomial Factoring

$$\large \textbf{Geometry}$$

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
Factorials
Rational Numbers

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

$$\large \textbf{Algebra}$$

Chapter	Wiki	Quiz
De Moivre's Theorem
Quadratic Discriminant
Linear Inequalities
Polynomial Inequalities
Absolute Value Inequalities
Rational Functions
Geometric Progressions
Binomial Theorem
Functional 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
Number Bases
Modular Arithmetic
Linear Diophantine Equations
Quadratic Diophantine Equations

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

$$\large \textbf{Algebra}$$

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

Here are six sample problems:

1. If $x$ is a positive real number, then find the minimum value of $$\frac{\big(x + \frac1x\big)^6 - \big(x^6 + \frac1{x^6}\big) - 2}{\big(x + \frac1x\big)^3 + \big(x^3 + \frac1{x^3}\big)}.$$

2. Suppose that $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively of $\triangle ABC.$ The perpendiculars to the sides $AB$ and $AC$ at $P$ and $Q$ respectively meet at $D,$ an interior point of $\triangle ABC.$ If $M$ is the midpoint of $BC,$ prove that $PM = QM$ if and only if $\angle BDP = \angle CDQ.$

3. Let $$N = 2^5 + 2^{5^2} + 2^{5^3} + \cdots + 2^{5^{2015}}.$$ Written in the usual decimal form, find the last two digits of the number $N.$

4. Two circles $\Sigma_1$ and $\Sigma_2$ having respective centers at $C_1$ and $C_2$ intersect at $A$ and $B.$ Let $P$ be a point on the segment $AB$ with $AP \neq PB.$ The line through $P$ perpendicular to $C_1P$ meets $\Sigma_1$ at $C$ and $D.$ The line through $P$ perpendicular to $C_2P$ meets $\Sigma_2$ at $E$ and $F.$ Prove that $C, D, E,$ and $F$ form a rectangle.

5. Solve the equation $$y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20$$ for positive integers $x$ and $y.$

6. From the list of positive integers, suppose we remove all multiples of 7, 11, and 13. At which position in the resulting list does the number 1002 appear? And what number occurs in position 3600?

Here are another six sample problems:

1. Let $ABC$ be a triangle. Let $B'$ and $C'$ denote respectively the reflection of $B$ and $C$ in the internal bisector of $\angle A$. Show that triangles $ABC$ and $AB'C'$ have the same incenter.

2. Let $P(x)= x^{2}+ax+b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $s≠t$ such that $P(s)= t$ and $P(t)= s$. Prove that $b-st$ is a root of the equation $x^{2}+ax+b-st=0$.

3. Find all integers $a,b,c$ such that $a^{2}=bc+1, b^{2}=ca+1$.

4. Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

5. Two circles $G_{1}$ and $G_{2}$ in a plane intersect at two points $A$ and $B$, and the center of $G_{2}$ lies on $G_{1}$. Let $C$ and $D$ be on $G_{1}$ and $G_{2}$, respectively, such that $C,B,$ and $D$ are collinear. Let $E$ on $G_{2}$ be such that $DE$ is parallel to $AC$. Show that $AE=AB$.

6. Find all real numbers $a$ such that $4<a<5$ and $a\big(a-3\{a\}\big)$ is an integer. $\big($Here $\{a\}$ denotes the fractional part of $a$. For example {$1.5$}= $0.5$; {$-3.4$}=$0.6.\big)$

Here are yet another six sample problems:

1. In an acute-angled $\triangle ABC$, $\angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect $AC$ at $X$ and $Y,$ respectively. Prove that circumcenter of $\triangle ABC$ is the incenter of $\triangle BXY.$

2. Let $x,y,z$ be positive real numbers. Prove that $$\frac { { y }^{ 2 }+{ z }^{ 2 } }{ x } +\frac { { z }^{ 2 }+{ x }^{ 2 } }{ y } +\frac { { x }^{ 2 }+{ y }^{ 2 } }{ z } \ge 2(x+y+z).$$

3. Find all pairs of $(x,y)$ of positive integers such that $2x+7y$ divides $7x+2y$.

4. For any positive integer $n>1,$ let $P(n)$ denote the largest prime not exceeding $n.$ Let $N(n)$ denote the next prime larger than $P(n)$. $\big($For example, $P(10)=7$ and $N(10)=11.\big)$ If $n$ is a prime number, prove that $$\frac { 1 }{ P(2)N(2) } +\frac { 1 }{ P(3)N(3) } +\cdots+\frac { 1 }{ P(n)N(n) } =\frac { n-1 }{ 2n+2 }.$$

5. Let $\triangle ABC$ be a triangle with $AB>AC$. Let $P$ be a point on the line beyond $A$ such that $AP+PC=AB$. Let $M$ be the midpoint of $BC$ and let $Q$ be a point on the side $AB$ such that $CQ\bot AM$. Prove that $BQ=2AP.$

6. Each square of an $n \times n$ grid $(n$ odd$)$ is arbitrarily filled with either by $1$ or by $-1$. Let ${ r }_{ j }$ and ${ c }_{ k }$ denote the product of all numbers in the $j^\text{th}$ row and the $k^\text{th}$ column, respectively, for $1\le j,k\le n$. Prove that $$\sum _{ j=1 }^{ n }{ { r }_{ j } } +\sum _{ k=1 }^{ n } c_ {k} \neq 0.$$

• Cauchy-Schwarz Inequality

• Applying the Cauchy-Schwarz Inequality