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# 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...

## 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.

## 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 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 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 $$0 < |f^{'}( \zeta )|< 1 \ \rightarrow$$...