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# I.S.I. & C.M.I. Entrance 2019 – Mock Tests and Review Classes

From advanced Number Theory to beautiful geometry and combinatorics. Prepare for some seriously interesting mathematics!

## Our Classes for I.S.I. & C.M.I. Entrance 2019

### I.S.I. and C.M.I. Entrance program review classes

Take five full-length I.S.I. Entrance Model Test (B.Stat – B.Math)

1. Attend the model tests online or at Calcutta Offline Center (near Tollygunge)
2. Attend review sessions on Number Theory, Geometry, Algebra and Combinatorics.
3. Mock Interview access (if the student qualifies the actual I.S.I. Entrance written test in May)

Access Fee: ₹ 5750

Alternatively you may join Online Classroom Program for I.S.I. Entrance 2020, and 2021.

## Shortest distance between curves – I.S.I. Entrance 2019 Subjective Solution Problem 8

Shortest path between two smooth curves is along the common normal. We use this fact to solve 8th problem of I.S.I. Entrance 2019 (UG Subjective)

## I.S.I. B.Stat and B.Math Entrance 2019 (UG) Solutions, Hints, and Answer Key

Indian Statistical Institute B.Stat, B.Math Entrance 2019 (UG Answer Key, Sequential Hints and more)

## Limit to Function – ISI UG 2019 Subj Problem 2

ISI BStat 2019, Subjective Problem 2 involving solving a limit to find the functional form of a function and its point of discontinuity by Sequential Hints.

## Sketching a complex set – I.S.I. Entrance 2019 Subjective Solution Problem 3

ISI Entrance 2019, Subjective Problem 3 involves sketching a set of complex numbers. We provide sequential hints that leads to solution.

## Locus of vertex of an equilateral triangle

A beautiful geometry problem from Math Olympiad program that involves locus of a moving point. Sequential hints will lead you toward solution.

## A Proof from my Book

This is proof from my book - my proof of my all-time favorite true result of nature - Pick's Theorem. This is the simplest proof I have seen without using any high pieces of machinery like Euler number as used in The Proofs from the Book. Given a simple polygon...

## ISI BStat 2018 Subjective Problem 2

Sequential Hints: Step 1: Draw the DIAGRAM with necessary Information, please! This will convert the whole problem into a picture form which is much easier to deal with. Step 2: Power of a Point - Just the similarity of $$\triangle QOS$$ and $$\triangle POR$$ By the...

## ISI BStat 2018 Subjective Problem 1

The solution will be posted in a sequential hint based format. You have to verify the steps of hints. Sequential Hints: Step 1: Solution set of sin($$\frac{x+y}{2}$$) = 0 is {$${x + y = 2n\pi : n \in \mathbb{N}}$$}- A set of parallel straight lines. The picture looks...

## A Math Conversation – I

Inspired by the book of Precalculus written in a dialogue format by L.V.Tarasov, I also wanted to express myself in a similar fashion when I found that the process of teaching and sharing knowledge in an easy way is nothing but the output of a lucid discussion between...

## The 3n+1 Problem

This problem is known as Collatz Conjecture. Consider the following operation on an arbitrary positive integer: If the number is even, divide it by two.If the number is odd, triple it and add one. The conjecture is that no matter what value of the starting number, the...

## The Dhaba Problem

Suppose on a highway, there is a Dhaba. Name it by Dhaba A. You are also planning to set up a new Dhaba. Where will you set up your Dhaba? Model this as a Mathematical Problem. This is an interesting and creative part of the BusinessoMath-man in you. You have to...

## The Organic Math of Origami

Did you know that there exists a whole set of seven axioms of Origami Geometry just like that of the Euclidean Geometry? The Origami in building the solar panel maximizing the input of Solar Power and minimizing the Volume of the satellite Instead of being very...

## Natural Geometry of Natural Numbers

Natural numbers also have a natural geometry of their. This post is about how they look in practice.

## Really understanding Barycentric coordinates

Sometimes we are interested in the relative position of a point with respect to a triangle. Is it close to the vertices? Is it closer to one of the sides compared to the other sides? This brings home the notion of mass point coordinates or barycentric coordinates.

## Problem Solving Marathon Week 2

We are having a full fledged Problem Solving Marathon. We are receiving wonderful responses from the end of our students which is making the session more and more alluring day by day. Here we are providing the problems and hints of "Problem Solving Marathon...

## Cheenta @ This Week

March 18 to 24January brings great news! Sambuddha got an offer from Cambridge. Aditya got multiple offers including one from Oxford. Soumyadeep cracked the TIFR entrance! We feel ecstatic. The entire credit goes to the students. We feel awesome to be a part of their...

## INMO 2019 Discussion

INMO is organized by HBCSE-TIFR. This post is dedicated for INMO 2019 Discussion. You can post your ideas here.

## Test of Mathematics Solution Objective 401 – Trigonometric Series

Summing a sequence of trigonometric ratios can be tricky. This problem from I.S.I. Entrance is an example.

## Understanding Simson Lines

Simson lines arise naturally. Imagine a triangle as a reference frame. Let a point float on the plane of the triangle. How far is the point from the sides of the triangle?

## What if a Simson Line moves!

A beautiful curved triangle appears when we run along the circumference! A magical journey into the geometry of Steiner’s Deltoid.

## 2016 ISI Objective Solution Problem 1

Problem The polynomial $$x^7+x^2+1$$ is divisible by (A) $$x^5-x^4+x^2-x+1$$ (B) $$x^5-x^4+x^2+1$$ (C) $$x^5+x^4+x^2+x+1$$ (D) $$x^5-x^4+x^2+x+1$$ . Also Visit: I.S.I. & C.M.I Entrance Program Understanding the Problem: The problem is easy...

## Test of Mathematics Solution Objective 398 – Complex Number and Binomial Theorem

Try a beautiful problem from complex numbers and geometry. It is from I.S.I. Entrance. We have created sequential hints to make this mathematical journey enjoyable!

## ISI – CMI entrance Book List

ISI – CMI entrance book list is useful for B.Stat and B.Math Entrance of Indian Statistical Institute, B.Sc. Math Entrance of Chennai Mathematical Institute

## Test of Mathematics Solution Objective 394 Power of Complex Number

Complex numbers and geometry are very closely related. We consider a problem from I.S.I. Entrance that uses this geometric character complex numbers.

## AM GM Inequality, Euler Number – Stories in Real Analysis

A.M.- G.M. Inequality can be used to prove the existence of Euler Number. A fascinating journey from classical inequalities to invention of one of the most important numbers in mathematics!

## Homework, Duality, Euler Number and Cheenta this week!

Hello mathematician! I do not like homework. They are boring ‘to do’ and infinitely more boring to ‘create and grade’. I would rather read Hilbert’s ‘Geometry and Imagination’ or Abanindranath’s ‘Khirer Putul’ at that time. Academy Award winner Michael...

## Lets invent Euler Number!

Do you want to invent new numbers and new functions? The story of how any age old banking formula led to the discovering of real analysis!

## Golden Ratio and Right Triangles – when geometry meets number theory

The golden ratio is arguably the third most interesting number in mathematics. We explore a beautiful problem connecting Number Theory and Geometry.

## I.S.I Entrance Solution – locus of a moving point

This is an I.S.I. Entrance Solution Problem: P is a variable point on a circle C and Q is a fixed point on the outside of C. R is a point in PQ dividing it in the ratio p:q, where p> 0 and q > 0 are fixed. Then the locus of R is (A) a circle; (B) an ellipse; ...

## I.S.I. Entrance Solution Sequence of isosceles triangles -2018 Problem 6

Let, $$a \geq b \geq c > 0$$ be real numbers such that for all natural number n, there exist triangles of side lengths $$a^n,b^n,c^n$$ Prove that the triangles are isosceles. If a, b, c are sides of a triangle, triangular inequality assures that difference of...

## Bases, Exponents and Role reversals (I.S.I. Entrance 2018 Problem 7 Discussion)

Let $$a, b, c$$ are natural numbers such that $$a^{2}+b^{2}=c^{2}$$ and $$c-b=1$$. Prove that (i) a is odd. (ii) b is divisible by 4 (iii) $$a^{b}+b^{a}$$ is divisible by c Notice that $$a^2 = c^2 - b^2 = (c+b)(c-b)$$ But c - b = 1. Hence $$a^2 = c + b$$. But c...

## Limit is Euler!

Let $$\{a_n\}_{n\ge 1}$$ be a sequence of real numbers such that $$a_n = \frac{1 + 2 + ... + (2n-1)}{n!} , n \ge 1$$ . Then $$\sum_{n \ge 1 } a_n$$ converges to ____________ Notice that $$1 + 3 + 5 + ... + (2n-1) = n^2$$. A quick way to remember this is sum of...

## Real Surds – Problem 2 Pre RMO 2017

Suppose $$a,b$$ are positive real numbers such that $$a\sqrt{a}+b\sqrt{b}=183$$. $$a\sqrt{b}+b\sqrt{a}=182$$. Find $$\frac{9}{5}(a+b)$$. This problem will use the following elementary algebraic identity: $$(x+y)^3 = x^3 + y^3 + 3x^2y + 3xy^2$$ Can you...

## Integers in a Triangle – AMC 10A

In this post we have discussed AMC 10A 2018 problem number 13.

## Tools in Geometry for Pre RMO, RMO and I.S.I. Entrance

Tools in Geometry is very useful for pre regional mathematical olympiad, regional mathematical olympiad as well as I.S.I. & C.M.I entrance.

## PreRMO and I.S.I. Entrance Open Seminar

Advanced Mathematics Seminar 2 hours An Open seminar for Pre-RMO and I.S.I. Entrance 2019 aspirants. We will work on topics from Number Theory, Geometry and Algebra. Registration is free. There are only 25 seats available. Date: 29th June, Friday, 6 PM Students...

## I.S.I. 2018 Problem 5 – a clever use of Mean Value Theorem

If a function is 'nice' (!), then for every secant line, there is a parallel tangent line! This is the intuition behind the Mean Value Theorem from Differentiable Calculus. The fifth problem from I.S.I. B.Stat and B.Math Entrance 2018, has a clever application of this...

## I.S.I 2018 Problem 4 Solution -Leibniz Rule

This is I.S.I 2018 Problem 4 Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I....