# 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

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### 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

## 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

There is an intuitive definition of perpendicularity. It does not involve angle. Instead, it involves the notion of distance. Consider a point P and a line L not passing through it. If you wish to walk from P to L, the path of shortest distance is the perpendicular...

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

Geometry is perhaps the most important topic in mathematics as far as Math Olympiad and I.S.I. Entrance goes. The following list of results may work as an elementary set of tools for handling some geometry problems. 'Learning' them won't do any good. One should 'find'...

## 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....

## Power of a Point – ISI 2018 Problem 2

The Problem Suppose that \(PQ\) and \(RS\) are two chords of a circle intersecting at a point \(O\). It is given that \(PO=3 \text{cm}\) and \(SO=4 \text{cm}\). Moreover, the area of the triangle \(POR\) is \(7 \text{cm}^2\). Find the area of the triangle \(QOS\). Key...

## 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...

## 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

The central goal of Combinatorics is to count things. Usually, there is a set of stuff that you would want to count. It could be number of permutations, number of seating arrangements, number of primes from 1 to 1 million and so on. Counting number of elements in a...

## 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 \(...

## Geodesics

How can we imagine 'straight lines' on a sphere? Any 'line' drawn on the surface of the sphere appears to be 'curved'. We must come up with some definition of 'straight-line' that allows 'curving'. This is actually simpler than it sounds. Let's declare the shortest...

## Euler Number in solids

Leonhard Euler was one of the greatest mathematician who ever lived. He lost his eyesight in the last few years of his life. That did not deter the Swiss genious to produce extraordinary mathematics. Consider the following geometric object. We...

## Geometry and Imagination – Open Seminar

What is it? An open-seminar on Geometry Who may attend? Math Olympiad, I.S.I., C.M.I. Entrance candidates and even College Students may attend. Date, Time, Venue 24th March 2018, Saturday 9 AM, online, worldwide 11 AM, at Cheenta Calcutta...

## Wheel of Numbers – Open Slate

This is a note from Cheenta Open Seminar - Wheel of Numbers. Rational Numbers are interesting in their own right. They are numbers that can be expressed as a ratio of two integers. $$ \frac {p}{q} $$ We will construct rational numbers by folding paper. Getting a...

## Not Pythagorean Triple

The Idea: Modular arithmetic provides a way to understand Pythagorean Equations. In the following videos we will explore the process. https://youtu.be/-gCq8l1p71I https://youtu.be/C7iOYkXrFRE ...

## Seemingly Easy, Yet intense!

Level 1 - Easy - 10 points The first problem is something which is somewhat elementary. From a biased coin(a coin where probability of heads is not be 1/2) how can you generate two events which are equally likely.(same probability). To restate it suppose there is a...

## Barycentric Coordinates – I.S.I 2014 Problem 2

The Setup: Points inside a triangle are 'related' algebraically to points at the vertices. We introduce the idea using a problem from I.S.I. B.Stat 2014. Problem: Let PQR be a triangle. Take a point A on or inside the triangle. Let f(x, y) = ax + by + c. Show...

## March Problem Lists

Understand Choose your course and download the Problem List document. The assignment page link is also added. You may submit the solutions there. This will constitute 50% of your monthly grade. Math Olympiad Early Bird (India) Number Theory Problem List (Early Bird)...

## A familiar Functional Equation

Cauchy's functional equations are very simple. The most familiar one has a simple formula: f(x + y) = f(x) + f(y) But first, for the uninitiated, what is a functional equation after all? What is a functional equation? Usually, functions appear as formulae. For...

## Contraction of a function – advanced Cheenta seminar

It is almost like deflating a balloon. But the effect is exponential. Today (29th January 2018, Monday), we have a special concept building cum problem-solving session on Contraction of a function. When 10 PM I.S.T. (29th January 2018) Where: Online (link...