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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.

## A Beautiful Equation using Positive Integers from Singapore Math Olympiad 2008

Understand the problemFind all pairs of positive integers  so that .Singapore MO 2008Number TheoryMediumAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!Note that $latex n+1$ has to be a prime, because any proper prime divisor of...

## As isoperimetric problem

Understand the problemShow that among all quadrilaterals of a given perimeter the square has the largest area.Indian National Mathematical Olympiad 1986GeometryEasyAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!Start with a...

## Does there exist a Magic Rectangle?

Magic Squares are infamous; so famous that even the number of letters on its Wikipedia Page is more than that of Mathematics itself. People hardly talk about Magic Rectangles. Ya, Magic Rectangles! Have you heard of it? No, right? Not me either! So, I set off to...

## An alluring trigonometric relation and its Implication

Understand the problemProve that a triangle  is right-angled if and only ifVietnam National Mathematical Olympiad 1981TrigonometryMediumChallenge and Thrill of Pre-college MathematicsStart with hintsDo you really need a hint? Try it first!Familiarity with the...

## I.S.I Entrance-2013 problem 2

This is the solution of ISI-2013 undergraduate entrance

## A function on squares

Understand the problem Let  be a real-valued function on the plane such that for every square  in the plane,  Does it follow that  for all points  in the plane?Putnam 2009 A1 Geometry Easy Mathematical Olympiad Challenges by Titu Andreescu Start with hintsDo you...

## Extremal Principle : I.S.I Entrance 2013 problem 4

This is a nice problem based on Well-ordering principle , from ISI entrance 2013

## Lattice point inside a triangle

Geometry problem from Iran math Olympiad .

## An isosceles triangle,on Trigonometry, I.S.I Entrance 2016, Solution to Subjective problem no. 6

Understand the problemLet $$a,b,c$$ be the sides of a triangle and $$A,B,C$$ be the angles opposite to those sides respectively. If $$\sin (A-B)=\frac{a}{a+b}\sin A\cos B-\frac{b}{a+b} \cos A \sin B$$, then prove that the triangle is isosceles. I.S.I. (Indian...

## I.S.I 2016 SUBJECTIVE PROBLEM – 1

Understand the problemSuppose that in a sports tournament featuring n players, each pairplays one game and there is always a winner and a loser (no draws).Show that the players can be arranged in an order P1, P2, . . . , Pn suchthat player Pi has beaten Pi+1 for all i...

## ISI 2019 : Problem #7

Understand the problem Let  be a polynomial with integer coefficients. Define and  for .If there exists a natural number  such that , then prove that either  or .   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2019. Subjective Problem...

## Working backward – C.M.I UG -2019

Understand the problem  If there exists a calculator with 12 buttons, 10 being the buttons for the digits and A and B being two buttons being processes where if n is displayed on the calculator if A is pressed it increases the displayed number by 1 and if B is pressed...

## Solution in Real – C.M.I -U.G-2019

Understand the problem .Find all real numbers x for which   C.M.I (Chennai mathematical institute ) U.G-2019   Algebra  8 out of 10.Start with hintsDo you really need a hint? Try it first!It is of the the form of   . Do you observe ?  where a=$$2^x$$ b=$$3^x$$...

## I.S.I 2019 Subjective Problem -4

Understand the problem Let  be a twice differentiable function such thatShow that there exist  such that  for all . I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2019. Subjective Problem no. 4 calculus  8.5 out of 10Problems In CALCULUS OF...

## Sum Of 1’S C.M.I UG-2019 Entrance

Understand the problem Find the sum 1+111+11111+1111111+.....1....111(2k+1) ones   C.M.I UG-2019 entrance examAlgebra 3.5 out of 10 challenges and trills of pre college mathematics    Start with hintsDo you really need a hint? Try it first!can you some how...

## C.M.I-2019 Geometry problem

Understand the problemlet O be a point inside a parallelogram ABCD such that $$\angle AOB+\angle COD =180$$ prove that $$\angle OBC =\angle ODC$$ C.M.I (Chennai mathematical institute UG-2019 entrance    Geometry 5 out of 10challenges and thrills of pre college...

## Triangle in complex plane – ISI 2019 Obj P8

This problem from ISI Entrance 2019 is an interesting application of complex numbers in geometry. Try your hands on this!

## Four Points on a Circle, ISI Entrance 2017, Subjective Problem no 2

Understand the problem Consider a circle of radius 6 as given in the diagram below. Let $$B,C,D$$ and $$E$$ be points on the circle such that $$BD$$ and $$CE$$, when extended, intersect at $$A$$. If $$AD$$ and $$AE$$ have length 5 and 4 respectively, and $$DBC$$ is a...

## The Product of Digits, ISI Entrance 2017, Subjective Solution to problem – 5.

Understand the problem  Let $$g : \mathbb{N} \to \mathbb{N}$$ with $$g(n)$$ being the product of digits of $$n$$.        (a) Prove that $$g(n)\le n$$ for all $$n \in \mathbb{N}$$ .        (b) Find all $$n \in \mathbb{N}$$ , for which $$n^2-12n+36=g(n)$$....

## Three Primes, ISI Entrance 2017, Subjective solution to Problem 6.

Understand the problemLet $$p_1,p_2,p_3$$  be primes with $$p_2\neq p_3$$, such that $$4+p_1p_2$$ and $$4+p_1p_3$$ are perfect squares. Find all possible values of $$p_1,p_2,p_3$$.   Start with hintsDo you really need a hint? Try it first!Let $$4+p_1p_2=m^2$$ and...

## System of n equations of Real Analysis , I.S.I Entrance 2008, Solution to Subjective Problem No. 9.

Understand the problem For $$n\ge3$$, determine all real solutions of the system of $$n$$ equations :                                                $$x_1+x_2+\cdots+x_{n-1}=\frac{1}{x_n}$$                                                                ...

## C.M.I. 2019 Entrance – Answer Key, Sequential Hints

CMI (Chennai Mathematical Institute) Entrance 2019, Sequential hints, answer key, solutions.

## A Trigonometric Substitution, ISI Entrance 2019, Subjective Solution to Problem – 6 .

Understand the problem For all natural numbers$$n$$, let          $$A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}$$           ($$n$$ many radicals) (a) Show that for $$n\ge 2, A_n=2\sin \frac{π}{2^{n+1}}$$. (b) Hence, or otherwise, evaluate the limit              ...

## Two Similar Triangles, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 2

Understand the problemSuppose that $$PQ$$ and $$RS$$ are two chords of a circle intersecting at a point $$O$$ , It is given that $$PO=3$$ cm and $$SO=4$$ cm . Moreover, the area of the triangle $$POR$$ is $$7 cm^2$$ . Find the are of the triangle $$QOS$$ .   I.S.I....

## Pythagorean Triple, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 7

Understand the problem Let $$a,b,c \in \mathbb{N}$$ be 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 $$c$$.   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

## Powers of 2 – I.S.I. B.Math/B.Stat Entrance 2019 Subjective Solution Problem 1

Understand the problem Prove that the positive integers $$n$$ that cannot be written as a sum of $$r$$ consecutive positive integers, with $$r>1$$ ,are of the form $$n=2^l$$ for some $$l\ge 0$$.   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

## Clocky Rotato Arithmetic

Do you know that CLOCKS add numbers in a different way than we do? Do you know that ROTATIONS can also behave as numbers and they have their own arithmetic? Well, this post is about how clocks add numbers and rotations behave like numbers. Consider the clock on earth....

## 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...