# I.S.I. & C.M.I. Entrance 2019 Test Series

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

### Choose One Test or Access All

### I.S.I. B.Stat – B.Math – One full length Model Test

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

- Attend the model test
**online or at Calcutta**Offline Center (near Tollygunge) - One full-length
**objective**test + One full-length**subjective**test - One online
**live discussion session**on the paper **Mock Interview**access (if the student qualifies the actual I.S.I. Entrance written test in May)

*Access Fee: ***₹ 750**

### I.S.I. & C.M.I. Entrance Test Series

Take **five** full-length I.S.I. Entrance Model Test (B.Stat – B.Math) and **two** full-length CMI Entrance Model Tests

- Attend the model tests
**online or at Calcutta**Offline Center (near Tollygunge) - Full-length
**objective**test + Full-length**subjective**test - Online
**live discussion**session on paper **Mock Interview**access (if the student qualifies the actual I.S.I. Entrance written test in May)- Access to
**Doubt Clearing**Forum

*Access Fee: ***₹ 3900**

Alternatively you may join Online Classroom Program

## 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

FebRuary 04 to 10January 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...

## 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.

## A Dream, An IMO 2018 Problem and A Why

IMO 2018 Problem 6 discussion is an attempt to interrogate our problem solving skill. This article is useful for the people who are willing to appear in any of the math olympiad entrances.

## 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....

## I.S.I 2018 Problem 3 – Functional Equation

This is a solution of I.S.I 2018 Problem 3 (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....

## I.S.I 2018 Problem 2 Discussion – Power of a Point

I.S.I 2018 Problem 2 Discussion is done based on the idea of ratio of areas of similar triangles is equal to ratio of squares on their corresponding sides.

## Solutions of equation – I.S.I. 2018 Problem 1

Find all pairs \( (x,y) \) with \(x,y\) real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Discussion: https://youtu.be/7Zx5n3nuGmo Back to Problems

## ISI Entrance Paper 2018 – B.Stat, B.Math Subjective

ISI Entrance Paper 2018 - from Indian Statistical Institute's B.Stat Entrance Also see: ISI and CMI Entrance Course at Cheenta Find all pairs ( (x,y) ) with (x,y) real, satisfying the equations $$sinbigg(frac{x+y}{2}bigg)=0~,~vert xvert+vert...

## Injection Principle – Combinatorics

Injection Principle is an very elegant idea to count objects. This idea is useful for olympiad students as well as for the I.S.I & C.M.I. students.

## Orthocenter and equal circles

Orthocenter (or the intersection point of altitudes) has an interesting construction. Take three equal circles, and make them pass through one point H. Their other point of intersection creates a triangle ABC. Turns out, H is the orthocenter of ABC. In this process,...

## Geometry of Motion: Open Seminar

Curving the infinity!Imagine squashing the infinite inside small circular disc! Lines bending or sliding to make room for the 'outside territory' inside. In the upcoming open slate Cheenta Seminar, we tackle this exciting problem from Geometry. Admission is free but...

## Bijections in Combinatorics (TOMATO Obj 168)

Bijection principle is a very useful tool for combinatorics. Here we pick up a problem that appeared in I.S.I.'s B.Stat-B.Math Entrance. Part 1: The problem and the hints https://youtu.be/EoGqTxQy940 Part 2 https://youtu.be/9gPEKehjxr8 Part 3...

## Algebraic Identity (TOMATO Objective 16)

Algebraic Identities can be tricky. Here we handle a simple case of repeated application of (a+b)(a-b). https://youtu.be/P3EXpj--Rbk

## Adventures in Geometry 1

Preface In geometry, transformation refers to the movement of objects. Adventures in Geometry 1 is the first part of "Adventures in Geometry" series.The content is presented as a relatively free-flowing dialogue between the Teacher and the Student. Also Visit: Math...

## Starters book in Algebra continued

Now lets discuss about the Second chapter named as SUBGROUPS . As mentioned before I am following the sequence of chapters from Herstein. IMPORTANT IDEAS: i) First go through the definition very well. You will see that H is a subgroup of G when H is a group under the...

## Shortest Path on Cube

An ant is sitting on the vertex of a cube. What is the shortest path along which it can crawl to the diagonally opposite vertex? The ant stays on the skin of the cube all the time. Here is a solution presented by the students in class: Open the cube (flatten it up)...

## Integer solutions of a three variable equation

Problem: Consider the following equation: \( (x-y)^2 + (y-z)^2 + (z - x)^2 = 2018 \). Find the integer solutions to this equation. Discussion: Set x - y = a, y - z = b. Then z - x = - (a+b). Clearly, we have, \( a^2 + b^2 + (-(a+b))^2 = 2018 \). Simplifying we have \(...