Mathematics Olympiad

Teacher

FractionsHub

Category:

INMO, Maths Olympiad, RMO

Online as well as Offline Classes are available

About the Course

Maths Olympiad for IOQM, RMO & INMO

We have two olympiad batches one is the advanced batch where we learn advanced topics which are taught in the IMO training camp and the other one is for the starters. We have our classroom facility where you will get all the lecture notes, class recordings, Assignments, Virtual Library, and much more. The online classes will be held online via Zoom and you will get a chance to meet the students who have actually cracked the exam and learn from them. The recordings of the classes will be posted after the class. For offline batches, you will have one offline class and one online class. We conduct one online class for the offline batch as we want you to get the exposure of the faculties who won’t be able to take the offline classes.

Demo Notes and Problems

Note: These are just sample notes. Complete lecture notes and materials will be given to the members during the course. Please stay tuned for more updates on sample notes.

How to enroll?

The course has been separated into two levels: Level 1 and Level 2. Level 1 and Level 2 will both consist of 6 months. You can click on “Get Course” and enroll for both levels at a cost of INR 12000 only or you may enroll for Level 1 now at a cost of INR 7500 only and then enroll for Level 2 at a cost of INR 7500 before it starts. No partial payments will be accepted. For other payment methods, like Google Pay, Bank Transfer, or UPI ID, you can WhatsApp at 9700803692. Registration will be closed once the limit has been reached. Registration will be based on first come first serve basics.

Success Stories

Aatman Supkar, INMO 2020

AMC-10-2020-2022

AMC-10-2020-2021-2022

AMC-12-2022-RMO-2018-2019-2020-2021

More than 20 RMO Qualified students at our INMOTC 2020 Camp

More than 50 Pre-RMO Qualified students at our RMO 2019 Camp

and much more…

Sample Problems

Number Theory Problems

Let $a_1, a_2, \cdots, a_{1000}$ be positive integers whose sum is $S$ . if $a_n! \bigg\vert n$ for $n = 1,2,3,...,1000$ , compute the maximum possible value of $S.$
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$
We say a natural number $n$ is perfect if the sum of all the positive divisors of $n$ is equal to $2n$ . For example, $6$ is perfect since its positive divisors $1,2,3,6$ add up to $12=2\times 6$ . Show that an odd perfect number has at least $3$ distinct prime divisors.
Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$
Let $a, m, n$ be positive integers. Prove that $\operatorname{gcd}\left(a^{m}-1, a^{n}-1\right)=a^{\operatorname{gcd}(m, n)}-1$
If $p$ is an odd prime, and $a, b$ are coprime, show that
$\quad \operatorname{gcd}\left(\frac{a^{p}+b^{p}}{a+b}, a+b\right) \in\{1, p\} .$
Are there distinct prime numbers $a, b, c$ which satisfy
$\quad a|b c+b+c, b| c a+c+a, c \mid a b+a+b ?$
Show that for all prime numbers $p$ ,
$\mathbb{Q}(p)=\prod_{k=1}^{p-1} k^{2k-p-1}$
is an integer.
Find all pairs $(a, b)$ of positive integers such that $a^{2017}+b$ is a multiple of $a b$ .
Let $a, b, c, d$ be pairwise distinct positive integers such that
$\quad \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$
is an integer. Prove that $a+b+c+d$ is not a prime number.

Combinatorics Problems

The numbers $1,2, \ldots, 100$ are arranged in the squares of a $10 \times 10$ table in the following way: the numbers $1, \ldots, 10$ are in the bottom row in increasing order, numbers $11, \ldots, 20$ are in the next row in increasing order, and so on. One can choose any number and two of its neighbors in two opposite directions (horizontal, vertical, or diagonal). Then either the number is increased by 2 and its neighbors are decreased by $1,$ or the number is decreased by 2 and its neighbors are increased by 1. After several such operations the table again contains all the numbers $1,2, \ldots, 100 .$ Prove that they are in the original order.
Find how many committees with a chairman can be chosen from a set of $n$ persons. Derive the identity
$\left(\begin{array}{l} n \\ 1 \end{array}\right)+2\left(\begin{array}{l} n \\ 2 \end{array}\right)+3\left(\begin{array}{l} n \\ 3 \end{array}\right)+\cdots+n\left(\begin{array}{l} n \\ n \end{array}\right)=n 2^{n-1} .$
Let $S$ be a set of $n$ persons such that:
(a) any person is acquainted with exactly $k$ other persons in $S$ ;
(b) any two persons that are acquainted have exactly $l$ common acquaintances in $S$
(c) any two persons that are not acquainted have exactly $m$ common acquaintances in $S$ .
Prove that

$m(n-k)-k(k-l)+k-m=0$
Let $n,p,q,r\in \mathbb{N},p,q,r\ge 0$ , such that $n\ge p+q+r$ .Then prove that :
${n\choose r}+{n-p-q\choose r}\ge {n-p\choose r}+{n-q\choose r}$
An even number of people are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Prove that at least two persons have the same of number persons between them as before the break
A set of numbers is called a sum-free set if no two of them add up to a member and if no member is the double of another member.
What is the maximum size of a sum-free subset of $S=\{1,2,...,2n+1\}$ ?
Find the maximum number of different integers that can be selected from the set $\{1,2,...,2013\}$ so that no two exist that their difference equals to $17$ .
In a billiard with shape of a rectangle $ABCD$ with $AB=2013$ and $AD=1000$ , a ball is launched along the line of the bisector of $\angle BAD$ . Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B
There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ( $A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$ ) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2$ tiles which can be used in such a tiling. Find all the values of $n$ .

Algebra Problems

For positive $p$ and $q$ such that $p+q=1$ , find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation
$\frac{f(x)-f(y)}{x-y}=f^{\prime}(p x+q y) \quad \text { for } \quad x, y \in \mathbb{R}, x \neq y$
Let a polynomial $P(x)=a_{m} x^{m}+a_{m-1} x^{m-1}+\cdots+a_{1} x+a_{0}$
with $a_{m}>0$ have $m$ distinct real zeros. Show that the polynomial $Q(x)=(P(x))^{2}-P^{\prime}(x)$ has
(a) exactly $m+1$ distinct real zeros if $m$ is odd,
(b) exactly $m$ distinct real zeros if $m$ is even.
(c) Prove that if a polynomial $P$ has $n$ distinct zeros greater than $1,$ then the polynomial

$Q(x)=\left(x^{2}+1\right) P(x) P^{\prime}(x)+x\left((P(x))^{2}+\left(P^{\prime}(x)\right)^{2}\right)$

has at least $2 n-1$ distinct real zeros.
Find all pairs $(x,y)$ of real numbers such that $|x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .
Show that
$\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16$

for all positive real numbers $a$ and $b$ such that $ab\geq 1$ .
Find all ordered triplets of $(x,y,z)$ real numbers that satisfy the following system of equation $x^3=\frac{z}{y}-\frac {2y}{z}$ $y^3=\frac{x}{z}-\frac{2z}{x}$ $z^3=\frac{y}{x}-\frac{2x}{y}$
Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.
Show that
$\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16$

for all positive real numbers $a$ and $b$ such that $ab\geq 1$ .
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$
Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$

Geometry Problems

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$ , ${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic.
Circles ${\omega_1}$ , ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and ${B}$ be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$
Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$ . The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$ . Let $D$ be the point of tangency of the $A$ -excircle of the triangle $ABC$ with the side $BC$ . Let $E$ be the reflection of $C$ with respect to the point $D$ . Prove that $PE\perp CQ$ .
A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$ , respectively. Prove that if $\angle BAC=60^{\circ}$ , then $AP+AQ+PQ<AB+AC+\frac{1}{2} BC$ .
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$ , respectively in a rectangle $ABCD$ . Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$ , respectively, and let $N$ be the intersection of the lines $PA$ and $QB$ . Let $X$ , $Y$ and $Z$ be the midpoints of the segments $AN$ , $KN$ and $AM$ , respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$ , $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$ . Prove that the lines $\ell_1$ , $\ell_2$ and $\ell_3$ are concurrent.
Let $ABC$ be an equilateral triangle , and $P$ be a point on the circumcircle of the triangle but distinct from $A$ , $B$ and $C$ . The lines through $P$ and parallel to $BC$ , $CA$ , $AB$ intersect the lines $CA$ , $AB$ , $BC$ at $M$ , $N$ and $Q$ respectively .Prove that $M$ , $N$ and $Q$ are collinear .
Let $ABC$ be an isosceles triangle with $AB=AC$ . Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$ . Prove that $HK \bot AB$ .
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ .
Let $MNPQ$ be a square of side length $1$ , and $A , B , C , D$ points on the sides $MN , NP , PQ$ and $QM$ respectively such that $AC \cdot BD=\frac{5}{4}$ . Can the set $\{AB , BC , CD , DA \}$ be partitioned into two subsets $S_1$ and $S_2$ of two elements each , so that each one has the sum of his elements a positive integer?
Consider a triangle $ABC$ with $\angle ACB=90^{\circ}$ . Let $F$ be the foot of the altitude from
$C$ . Circle $\omega$ touches the line segment $FB$ at point $P$ , the altitude $CF$ at point $Q$ and the
circumcircle of $ABC$ at point $R$ . Prove that points $A, Q, R$ are collinear and $AP = AC$ .