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.
Contents
Contest Information
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.
Basic Topics
These are the basic topics covered by the RMO. This is a great place to start learning if you’re new to the RMO!
\[\large \textbf{Algebra} \]
\[\large \textbf{Geometry} \]
\[\large \textbf{Combinatorics (Counting and Probability)} \]
Chapter Wiki Quiz Set Operations 
Principle of Inclusion/Exclusion Understanding Data
\[\large \textbf{Number Theory} \]
Intermediate Topics
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.
\[\large \textbf{Algebra} \]
\[\large \textbf{Geometry} \]
\[\large \textbf{Combinatorics (Counting and Probability)} \]
Chapter Wiki Quiz Rules of Sum and Product Discrete Probability Expected Value Combinations
\[\large \textbf{Number Theory} \]
Advanced Topics
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!
\[\large \textbf{Algebra} \]
\[\large \textbf{Geometry} \]
\[\large \textbf{Combinatorics (Counting and Probability)} \]
Chapter Wiki Quiz Geometric Probability Binomial Distribution Conditional Probability
\[\large \textbf{Number Theory} \]
Sample Problems
Here are six sample problems:
If \(x\) is a positive real number, then find the minimum value of \[\frac{\big(x + \frac1x\big)^6 - \big(x^6 + \frac1{x^6}\big) - 2}{\big(x + \frac1x\big)^3 + \big(x^3 + \frac1{x^3}\big)}.\]
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 \(\angle BDP = \angle CDQ.\)
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 \(N.\)
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.
Solve the equation \[y^3 + 3y^2 + 3y = x^3 + 5x^2 - 19x + 20\] for positive integers \(x\) and \(y.\)
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:
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.
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 \(x^{2}+ax+b-st=0\).
Find all integers \(a,b,c\) such that \(a^{2}=bc+1, b^{2}=ca+1\).
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?
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 \(AE=AB\).
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:
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 \(\triangle BXY.\)
Let \(x,y,z\) be positive real numbers. Prove that \[\frac { { y }^{ 2 }+{ z }^{ 2 } }{ x } +\frac { { z }^{ 2 }+{ x }^{ 2 } }{ y } +\frac { { x }^{ 2 }+{ y }^{ 2 } }{ z } \ge 2(x+y+z).\]
Find all pairs of \((x,y)\) of positive integers such that \(2x+7y\) divides \(7x+2y\).
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 \[\frac { 1 }{ P(2)N(2) } +\frac { 1 }{ P(3)N(3) } +\cdots+\frac { 1 }{ P(n)N(n) } =\frac { n-1 }{ 2n+2 }.\]
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 \(BQ=2AP.\)
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 \[\sum _{ j=1 }^{ n }{ { r }_{ j } } +\sum _{ k=1 }^{ n } c_ {k} \neq 0. \]
Also Solve These for Practice
Let \(a\), \(b\), and \(c\) be positive real numbers such that
\[ \frac{a}{1+b} + \frac{b}{1 + c} + \frac{c}{1 + a} = 1.\]
What is the maximum value of \(abc?\)
Let \(P(x)\) be a polynomial of degree 2015 which satisfies \(P(k) = 2^{k}\) for all \(k=0,1,2,3, \ldots, 2015\). If the value of \(P(2016)\) can be expressed as
\[a^{b} - c,\]
where \(a\) and \(c\) are coprime integers and \(a\) is a prime, find the value of \(a+b+c\).
Try this set RMO Practice Problems.
\[ \begin{cases} \begin{align} (1 + x)\big(1 + x^2\big)\big(1 + x^4\big) &= 1 + y^7 \\\\ (1 + y)\big(1 + y^2\big)\big(1 + y^4\big) &= 1 + x^7 \end{align} \end{cases}\]
Find the number of ordered pairs of real numbers \((x, y)\) for which the above system of equations is satisfied.
Let \(F(x) = x - \lfloor x \rfloor \). Find the number of solutions to the equation \(F(x) + \dfrac1{F(x)} = 1 \).
\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.
Find the last two digits of \(17^{28}\).
Find the coefficient of \(a^5b^5c^5d^6\) in the expansion of \((bcd+acd + abd + abc)^7.\)
\[\large \frac 1{\lfloor x \rfloor} + \frac 1{\lfloor 2 x \rfloor} = \{ 9x \} + \frac 13 \]
If the smallest positive solution of \(x\) satisfying the equation above is of the form \(\frac {m}{n}\), where \(m\) and \(n\) are coprime positive integers, find \(m-n\).
\[\] Notations:
- \( \lfloor \cdot \rfloor \) denotes the floor function.
- \( \{ \cdot \} \) denotes the fractional part function.
Find the remainder when \(2^{2014}\) is divided by 1000.
Find the number of pairs of integers \(x\) and \(y\) that satisfy the equation
\[ x^2=2y^2-8y+3 . \]
Find the smallest positive integer \(N\) such that \( 13^N \equiv 1 \pmod{2013}\).
\(\)
Details and Assumptions:
- You may choose to refer to the modulo arithmetic notation.
- 0 is not a positive integer.
Let \(a\) and \(b\) be relatively prime positive integers. Find the sum of all possible values of
\[\gcd \left(\frac{a^{2017} + b^{2017}}{a + b}, \ a + b \right).\]
Note: 2017 is prime.
Determine the number of pairs \((x,y)\) of integers such that
\[ 1+2^x+2^{2x+1}=y^2.\]
Source: an IMO problem
Find the domain of \(x\) satisfying
\[ \Big\lfloor x + \big\lfloor x + \lfloor x \rfloor \big\rfloor \Big\rfloor + 6= 0. \]
\(\)
Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.
How many fractions are there such that they can be written simultaneously in the forms
\[\dfrac { 7k-5 }{ 5k-3 } \quad \text{and} \quad \dfrac { 6l-1 }{ 4l-3 } \]
for some integers \(k\) and \(l?\)
Let \(f(x) = \Large{x^{x^{x^x}}}\). Find the last two digits of \(f(17) + f(18) + f(19) + f(20)\).