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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.

## Solving a congruence

Understand the problemProve that the number of ordered triples in the set of residues of $latex p$ such that , where and is prime is . Brazilian Olympiad Revenge 2010 Number Theory Medium Elementary Number Theory by David Burton Start with hintsDo you really need...

## Inequality involving sides of a triangle

Understand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

## Vectors of prime length

Understand the problemGiven a prime number and let be distinct vectors of length with integer coordinates in an Cartesian coordinate system. Suppose that for any , there exists an integer such that all three coordinates of is divisible by . Prove that .Kürschák...

## Missing digits of 34!

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

## An inequality involving unknown polynomials

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

## Hidden triangular inequality (PRMO Problem 23, 2019)

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

## Bangladesh MO 2019 Problem 1 – Number Theory

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

## Functional equation dependent on a constant

A basic and beautiful application of Numebr Theory and Modular Arithmetic to the Bangladesh MO 2019 Problem 1.

## Pigeonhole principle exercise

Understand the problemFind all real numbers for which there exists a non-constant function satisfying the following two equations for all i) andii) Baltic Way 2016 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need...

## An interesting arithmetic function on Number Theory using the Prime Numbers

Understand the problemWe are given sets of size each. The size of the intersection of any two sets is exactly . Prove that all the sets have a common element.Austrian-polish mathematical Olympiad 1978 Combinatorics Easy An Excursion in Mathematics Start with hintsDo...

## Belarus MO 2018 Problem 10.5 – Number Theory

Understand the problemLet be the set of positive integers. Determine all functions such that is divisible by for all positive integers .APMO 2019 Number Theory Medium An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Considering...

## machine learning course offered by Cheenta

This problem is a cute and simple application of the technique of reducing the sample space of solutions by the modulo operation.

## Lottery tickets

this course is totally free. the links of prev. classes can be obtained from Ms. Shabana of Cheenta (admin). For any queries / comments / suggestion you can contact : prabirdg0@gamil.com ML-course-covered-so-farDownload

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

Understand the problemWe are given sets of size each. The size of the intersection of any two sets is exactly . Prove that all the sets have a common element.Austrian-polish mathematical Olympiad 1978 Combinatorics Easy An Excursion in Mathematics Start with hintsDo...

## As isoperimetric problem

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...

## Does there exist a Magic Rectangle?

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...

## An alluring trigonometric relation and its Implication

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...

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

Understand the problemFind all the polynomials of a degree with real non-negative coefficients such that , . Albanian BMO TST 2009 Algebra Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!This problem is all about...

## A function on squares

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

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

Understand the problem34!=295232799cd96041408476186096435ab000000 Find $latex a,b,c,d$ (all single digits).BMO 2002 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!Get prepared to find the residue of 34! modulo...

## Lattice point inside a triangle

Understand the problemLet be the lengths of sides of a (possibly degenerate) triangle. Prove the inequalityLet be the lengths of sides of a triangle. Prove the inequalityCaucasus Mathematical Olympiad Inequalities Easy An Excursion in Mathematics Start with hintsDo...

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

This problem is a cute and simple application of the technique of reducing the sample space of solutions by the modulo operation.

## I.S.I 2016 SUBJECTIVE PROBLEM – 1

Problem Let ABCD be a convex cyclic quadrilateral . Suppose P is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of ABCD is the least .If {PA,PB,PC,PD} = {3,4,6,8}.What is the maximum possible area of ABCD? TopicGeometry...

## ISI 2019 : Problem #7

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...

## Working backward – C.M.I UG -2019

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...

## Solution in Real – C.M.I -U.G-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...

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

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...

## Sum Of 1’S C.M.I UG-2019 Entrance

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...

## C.M.I-2019 Geometry problem

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...

## Triangle in complex plane – ISI 2019 Obj P8

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...

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

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...

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

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...

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

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 $$n^2-12n+36=g(n)$$....

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

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...

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

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 $$n^2-12n+36=g(n)$$....

## A Trigonometric Substitution, ISI Entrance 2019, 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...

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

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 ...

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

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....

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

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 ...

## Clocky Rotato Arithmetic

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....