# You will need pen and paper

# I.S.I. Entrance Problems

## INMO 2020 Problems, Solutions and Hints

INMO 2020 (Indian National Math Olympiad) 2020 Problems and Solutions. We provide sequential hints so that you can try the problems on your own.

## INMO 2020 Problem 4

Let $latex n\ge 3$ be an integer and $latex a_1,a_2,\cdots a_n$ be real numbers satisfying $latex 1<a_2\le a_2\le a_3\cdots \le a_n$. If $latex \Sigma_ia_i=2n$ then prove that $latex 2+a_1+a_1a_2+a_1a_2a_3+\cdots +a_1a_2\cdots a_{n-1}\le a_1a_2\cdots a_n$.The...

## Beautiful problems from Coordinate Geometry

The following problems are collected from a variety of Math Olympiads and mathematics contests like I.S.I. and C.M.I. Entrances. They can be solved using elementary coordinate geometry and a bit of ingenuity.

## An interesting biquadratic from ISI Entrance 2005

How to combine algebra and geometry to solve a biquadratic? Try this beautiful problem from ISI Entrance 2005. We provide knowledge graph and video.

## cos(sin(x)) function in ISI Entrance

A simple trigonometric equation from ISI Entrance. Try this problem. We also added a quiz, some related problems, and finally video.

## Geometry of AM GM Inequality

AM GM Inequality has a geometric interpretation. Watch the video discussion on it and try some hint problems to sharpen your skills.

## Counting triangles in ISI Entrance

Can you combine geometry and combinatorics? This ISI Entrance problems requires just that. We provide sequential hints, additional problems and video.

## Paper folding geometry in ISI Entrance

A problem from ISI Entrance that requires Paper folding geometry. We provide sequential hints so that you can try the problem!

## An Hour of Beautiful Proofs

Every week we dedicate an hour to Beautiful Mathematics - the Mathematics that shows us how Beautiful is our Intellect. This week, I decided to do three beautiful proofs in this one-hour session... Proof of Fermat's Little Theorem ( via Combinatorics )It uses...

# Math Olympiad

## A trigonometric polynomial ( INMO 2020 Problem 2)

Indian National Math Olympiad (INMO 2020) Solution and sequential hints to problem 2

## Kites in Geometry | INMO 2020 Problem 1

Try this beautiful geometry problem from INMO (Indian National Math Olympiad) 2020). We provide solution with sequential hints so that you can try.

## Geometry of AM GM Inequality

AM GM Inequality has a geometric interpretation. Watch the video discussion on it and try some hint problems to sharpen your skills.

## Geometry of Cauchy Schwarz Inequality

Cauchy Schwarz Inequality is a powerful tool in Algebra. However it also has a geometric meaning. We provide video and problem sequence to explore that.

## RMO 2019 Maharashtra and Goa Problem 2 Geometry

Understand the problemGiven a circle $latex \Gamma$, let $latex P$ be a point in its interior, and let $latex l$ be a line passing through $latex P$. Construct with proof using a ruler and compass, all circles which pass through $latex P$, are tangent to $latex...

## RMO 2019 (Maharashtra Goa) Adding GCDs

Can you add GCDs? This problem from RMO 2019 (Maharashtra region) has a beautiful solution. We also give some bonus questions for you to try.

## Number Theory, Ireland MO 2018, Problem 9

This problem in number theory is an elegant applications of the ideas of quadratic and cubic residues of a number. Try with our sequential hints.

## Number Theory, France IMO TST 2012, Problem 3

This problem is an advanced number theory problem using the ideas of lifting the exponents. Try with our sequential hints.

## Algebra, Austria MO 2016, Problem 4

This algebra problem is an elegant application of culminating the ideas of polynomials to give a simple proof of an inequality. Try with our sequential hints.

# College mathematics

## Supremum and Infimum: IIT JAM 2018 Problem 11

This is a problem which appeared in IIT JAM 2018. So this problem requires basic concepets of supremum and infimum from real analysis part.

## Sequences & Subsequences : IIT 2018 Problem 10

This problem appeared in IIT JAM 2018 whch pricisely reqiures concepts of sequences and subsequences from mathematical field real analysis

## Cyclic Groups & Subgroups : IIT 2018 Problem 1

This is an application abstract algebra question that appeared in IIT JAM 2018. The concept required is the cyclic groups , subgroups and proper subgroups.

## Acute angles between surfaces: IIT JAM 2018 Qn 6

This is an application analysis question that appeared in IIT JAM 2018. The concept required is the multivarible calculus and vector analysis.

## Finding Tangent plane: IIT JAM 2018 problem 5

What are we learning?Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 will use these ideasUnderstand the problemThe tangent plane to the surface $latex z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by \(x-3y+z=0\)...

## An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

## Linear Algebra total recall (Open Seminar)

Open Seminar on linear algebra. A review of all major ideas. Even if you have little or no knowledge about Linear Algebra, you may join. Register now.

## 4 questions from Sylow’s theorem: Qn 4

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

## 4 questions from Sylow’s theorem: Qn 3

I have prepared some common questions ok application of Sylow’s theorem with higher difficulty level. It is most propably cover all possible combination.

# Research Tracks

## Arithmetical Dynamics: Part 6

Consider fix point of \( R(z) = z^2 - z \) . Which is the solution of $$ R(z) = z \\ \Rightarrow z^2 - z =z \\ \Rightarrow z^2 - 2z =0 \\ \Rightarrow z(z -2) =0 $$ Now , consider the fix point of \( R^2(z) \) . \( \\ \) $$ i.e. R^2(z) = R . R(z) = R(z^2 -z)\\...

## Arithmetical Dynamics: Part 5

And suppose that R has no periodic points of period n . Then (d, n) is one of the pairs \( (2,2) ,(2,3) ,(3,2) ,(4 ,2) \) , each such pair does arise from some R in this way . The example of such pair is $$ 1. R(z) = z +\frac {(w-1)(z^2 -1)}{2z} ; it \ has \ no \...

## Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...

## Arithmetical Dynamics: Part 4

\( P^m(z) = z \ and \ P^N(z)=z \ where \ m|N \Rightarrow (P^m(z) - z) | (P^N(z)-z) \) The proof of the theorem in Part 0 : Let , P be the polynomials satisfying the hypothesis of theorem 6.2.1 . Let , \( K = \{ z \in C | P^N(z) =z \} \\ \) and let \( M =\{ m \in Z : 1...

## Arithmetical Dynamics: Part 3

Theory: Let \( \{ \zeta_1 , ......., \zeta_m \} \) be a ratinally indifferent cycle for R and let the multiplier of \( R^m \) at each point of the cycle be \( exp \frac {2 \pi i r}{q} \) where \( (r,q) =1 \) . Then \( \exists \ k \in Z \) and\( mkq \) distinct...

## Arithmetical Dynamics: Part 2

The lower bound calculation is easy . But for the upper bound , observe that each \( z \in K \) lies in some cycle of length m(z) and we these cycles by \( C_1 , C_2 .....,C_q \) . Further , we denote the length of the cycle by \( m_j \) , so , if \( z \in C_j \) then...

## Arithmetical Dynamics: Part 1

Definition: Suppose that \( \zeta \in C \) is a fixed point of an analytic function \( f \) . Then \( \zeta \) is : a) Super attracting if \( f^{'} (\zeta) =0 \rightarrow \) critical point of \( f \) b) Attractting if \( 0 < |f^{'}( \zeta )|< 1 \ \rightarrow \)...

## Research for School

Research projects for school students, in mathematics and data science. For advanced learners who are in love with mathematical science.

## Arithmetical Dynamics: Two possible problems

1.1. Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s theorem. Related reading: (1) Silverman Arithmetic of Dynamical Systems: p 165. Bifurcation polynomials. Exercise 4.12 outlines some known properties and open questions (** = open...