Workshop for I.S.I. and C.M.I. Entrance Exam 2022

About the Workshop

A 10-WEEKS CRASH COURSE for ISI & CMI entrance exam 2022.

This workshop is being organized for the I.S.I. and C.M.I. aspirants for B.Math, B.Stat (I.S.I.), and B.Sc (C.M.I.) entrance exams. This workshop will focus mainly on problem-solving. We have collected a wide variety of problems which we will be discussing in these ten weeks. We have very carefully chosen the problems such that they will be of great help to the students who are going to appear for the entrance exams this year or are preparing for it. The main topics to follow are Number Theory, Combinatorics, Polynomials, Geometry, Calculus e.t.c. and the problem sheets will be provided to the students beforehand so that they can try them at home before we discuss them in the class. We will also talk about the things you need to follow just before your exams and also those who are preparing for the exam will get to know all the major sources that they need to follow to enhance their preparation. The classes will be held online via Zoom and you will get a chance to meet the students who have actually cracked the exam and learn from them. The recordings of the classes will be posted after the class. The workshop will start on 15 th February 2022.

Starting 15 th February, 2022

Demo Notes and Problems

Note: These are just sample notes. Complete lecture notes and materials will be given to the members during the course. Please stay tuned for more updates on sample notes.

How to enroll?

The course has been separated into two levels: Level 1 and Level 2. Level 1 will begin on February 15th, 2022, and will last for five weeks and then Level 2 will resume and continue for 5 weeks. You can click on “Get Course” and enroll for both levels at a cost of INR 5000 only or you may enroll for Level 1 now at a cost of INR 3000 only and then enroll for Level 2 at a cost of INR 3000 before it starts. If one wishes to appear only for the Mock Tests then there is a separate plan for them which costs INR 1000 only. Please note that students who opt for the full course or opt for Level 1 and then Level 2 will not have to purchase the Mock Tests plan separately as it is already included in the course. You will only be able to register once you have paid the full amount. No partial payments will be accepted. For other payment methods, you can WhatsApp at 9700803692. Registration will be closed once the limit has been reached. Registration will be based on first come first serve basics.

Past year Question Papers, Solutions and Discussion

For ISI Previous Years B.Stat & B.Math: Click Here

For CMI Previous Years Undergraduate Mathematics Question papers and official solutions: Click Here

2021 Success Stories

  1. Manan Roy Choudhury, ISI B.Stat 2021

  2. Ankit Gayen, ISI B.Math 2021, CMI UG 2021

  3. Sevantee Basu, ISI B.Stat 2021

  4. Ritabrata Bhattacharyya, CMI UG 2021

  5. Deepta Basak, ISI B.Math 2021

Course Plan

The remaining three weeks will be planned according to the demands of the students.
Class Timings will be mostly from 8 to 10 pm or 9 to 11 pm.

Sample Problems

Number Theory Problems

  • Let a_1, a_2, \cdots, a_{1000} be positive integers whose sum is S. if a_n! \bigg\vert n for n = 1,2,3,...,1000, compute the maximum possible value of S.
  • Find all primes p and q such that 3p^{q-1}+1 divides 11^p+17^p
  • We say a natural number n is perfect if the sum of all the positive divisors of n is equal to 2n. For example, 6 is perfect since its positive divisors 1,2,3,6 add up to 12=2\times 6. Show that an odd perfect number has at least 3 distinct prime divisors.
    Note: It is still not known whether odd perfect numbers exist. So assume such a number is there and prove the result.
  • Find all triples (p, x, y) consisting of a prime number p and two positive integers x and y such that x^{p -1} + y and x + y^ {p -1} are both powers of p
  • Let a, m, n be positive integers. Prove that \operatorname{gcd}\left(a^{m}-1, a^{n}-1\right)=a^{\operatorname{gcd}(m, n)}-1
  • If p is an odd prime, and a, b are coprime, show that
    \quad \operatorname{gcd}\left(\frac{a^{p}+b^{p}}{a+b}, a+b\right) \in\{1, p\} .
  • Are there distinct prime numbers a, b, c which satisfy
    \quad a|b c+b+c, b| c a+c+a, c \mid a b+a+b ?
  • Show that for all prime numbers p,
    \mathbb{Q}(p)=\prod_{k=1}^{p-1} k^{2k-p-1}
    is an integer.
  • Find all pairs (a, b) of positive integers such that a^{2017}+b is a multiple of a b.
  • Let a, b, c, d be pairwise distinct positive integers such that
    \quad \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}
    is an integer. Prove that a+b+c+d is not a prime number.

Combinatorics Problems

  • The numbers 1,2, \ldots, 100 are arranged in the squares of a 10 \times 10 table in the following way: the numbers 1, \ldots, 10 are in the bottom row in increasing order, numbers 11, \ldots, 20 are in the next row in increasing order, and so on. One can choose any number and two of its neighbors in two opposite directions (horizontal, vertical, or diagonal). Then either the number is increased by 2 and its neighbors are decreased by 1, or the number is decreased by 2 and its neighbors are increased by 1. After several such operations the table again contains all the numbers 1,2, \ldots, 100 . Prove that they are in the original order.
  • Find how many committees with a chairman can be chosen from a set of n persons. Derive the identity

        \[\left(\begin{array}{l} n \\ 1 \end{array}\right)+2\left(\begin{array}{l} n \\ 2 \end{array}\right)+3\left(\begin{array}{l} n \\ 3 \end{array}\right)+\cdots+n\left(\begin{array}{l} n \\ n \end{array}\right)=n 2^{n-1} .\]

  • Let S be a set of n persons such that:
    (a) any person is acquainted with exactly k other persons in S;
    (b) any two persons that are acquainted have exactly l common acquaintances in S
    (c) any two persons that are not acquainted have exactly m common acquaintances in S.
    Prove that

        \[m(n-k)-k(k-l)+k-m=0\]

  • Let n,p,q,r\in \mathbb{N},p,q,r\ge 0 , such that n\ge p+q+r.Then prove that :
    {n\choose r}+{n-p-q\choose r}\ge {n-p\choose r}+{n-q\choose r}
  • An even number of people are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Prove that at least two persons have the same of number persons between them as before the break
  • A set of numbers is called a sum-free set if no two of them add up to a member and if no member is the double of another member.
    What is the maximum size of a sum-free subset ofS=\{1,2,...,2n+1\}?
  • Find the maximum number of different integers that can be selected from the set \{1,2,...,2013\} so that no two exist that their difference equals to 17.
  • In a billiard with shape of a rectangle ABCD with AB=2013 and AD=1000, a ball is launched along the line of the  bisector of \angle BAD. Supposing that the ball is reflected on the sides  with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B
  • There are two piles of coins, each containing 2010 pieces. Two players A and B play a game taking turns (A plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
  • A 9\times 7 rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with 90^\circ) and square tiles composed by four unit squares.
    Let n\ge 0 be the number of the 2 \times 2 tiles which can be used in such a tiling. Find all the values of n.

Calculus Problems

  • Suppose that f:\mathbb{R}\rightarrow[-1,1] is in C^{2} on \mathbb{R}, and it is given that:

        \[(f(0))^{2}+\left(f^{"}(0)\right)^{2}=4.\]

    Prove that there exists x_{0} \in \mathbb{R} such that f\left(x_{0}\right)+f^{''}\left(x_{0}\right)=0

  • Let f be continuous on [0,1] and differentiable on (0,1) . Suppose that f(0)=f(1)=0 and that there is x_{0} \in(0,1) such that f\left(x_{0}\right)=1 . Prove that \left|f^{\prime}(c)\right|>2 for some c \in(0,1).
  • Let f be differentiable continuous on [0,1] satisfies:
    \int_0^1 f(x)dx=5/2 ,
    \int_0^1 x.f(x)dx=3/2
    Prove that there exists c \in (0,1), such that f'(c) = 3.
    \item Compute the integral

        \[I=\int_{0}^{1} \sqrt[3]{2 x^{3}-3 x^{2}-x+1} d x\]

  • Suppose that f is differentiable on [0,1] and that f(0)= f(1)=0 . Suppose, further, that f^{\prime \prime} exists on (0,1) and is bounded (say \left|f^{\prime \prime}(x)\right| \leq A for \left.x \in(0,1)\right). Prove that

        \[\left|f^{\prime}(x)\right| \leq \frac{A}{2} \text { for } x \in[0,1]\]

    \item Find the antiderivatives of the function f:[0,2] \rightarrow \mathbb{R}

        \[f(x)=\sqrt{x^{3}+2-2 \sqrt{x^{3}+1}}+\sqrt{x^{3}+10-6 \sqrt{x^{3}+1}}\]

  • For nonzero a_{1}, a_{2}, \ldots, a_{n} and for \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} such that \alpha_{i} \neq \alpha_{j} for i \neq j, prove that the equation

        \[a_{1} x^{\alpha_{1}}+a_{2} x^{\alpha_{2}}+\cdots+a_{n} x^{\alpha_{n}}=0, \quad x \in(0, \infty)\]

    has at most n-1 roots in (0, \infty)

  • Compute the integral

        \[\int\left(1+2 x^{2}\right) e^{x^{2}} d x\]

  • Compute the integral

        \[\int_{0}^{a} \frac{d x}{x+\sqrt{a^{2}-x^{2}}} \quad(a>0)\]

  • Let f be infinitely differentiable on [0,1], and suppose that for each x \in[0,1] there is an integer n(x) such that f^{(n(x))}(x)=0 Prove that f coincides on [0,1] with some polynomial.

Algebra Problems

  • For positive p and q such that p+q=1, find all functions f: \mathbb{R} \rightarrow \mathbb{R} satisfying the equation

        \[\frac{f(x)-f(y)}{x-y}=f^{\prime}(p x+q y) \quad \text { for } \quad x, y \in \mathbb{R}, x \neq y\]

  • Let a polynomial P(x)=a_{m} x^{m}+a_{m-1} x^{m-1}+\cdots+a_{1} x+a_{0}
    with a_{m}>0 have m distinct real zeros. Show that the polynomial Q(x)=(P(x))^{2}-P^{\prime}(x) has
    (a) exactly m+1 distinct real zeros if m is odd,
    (b) exactly m distinct real zeros if m is even.
    (c) Prove that if a polynomial P has n distinct zeros greater than 1, then the polynomial

        \[Q(x)=\left(x^{2}+1\right) P(x) P^{\prime}(x)+x\left((P(x))^{2}+\left(P^{\prime}(x)\right)^{2}\right)\]

    has at least 2 n-1 distinct real zeros.

  • Find all pairs (x,y) of real numbers such that |x|+ |y|=1340 and x^{3}+y^{3}+2010xy= 670^{3} .
  • Show that

        \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]

    for all positive real numbers a and b such  that ab\geq 1.

  • Find all ordered triplets of (x,y,z) real numbers that satisfy the following system of equation x^3=\frac{z}{y}-\frac {2y}{z} y^3=\frac{x}{z}-\frac{2z}{x} z^3=\frac{y}{x}-\frac{2x}{y}
  • Find the maximum value of |\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}| where x is a real number.
  • Show that

        \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]

    for all positive real numbers a and b such  that ab\geq 1.

  • Find all triples (a,b,c) of real numbers such that the following system holds:

        \[\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}\]

  • Solve in positive real numbers: n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014

Geometry Problems

  • Let {AB} be a diameter of a circle  {\omega} and center {O} ,  {OC} a radius of {\omega} perpendicular to AB,{M} be a point of the segment \left( OC \right) . Let {N} be the second intersection point of line {AM} with {\omega} and {P} the intersection point of the tangents of {\omega} at points {N} and {B.} Prove that points {M,O,P,N} are cocyclic.
  • Circles {\omega_1} , {\omega_2} are externally tangent at point M and tangent internally with circle {\omega_3} at points {K} and L respectively. Let {A} and  {B}be the points that their common tangent at point {M} of circles {\omega_1} and {\omega_2} intersect with circle {\omega_3.} Prove that if {\angle KAB=\angle LAB} then the segment {AB} is diameter of circle {\omega_3.}
  • Let I be the incenter and AB the shortest side of the triangle ABC. The circle centered at I passing through C intersects the ray AB in P and the ray BA in Q. Let D be the point of tangency of the A-excircle of the triangle ABC with the side BC. Let E be the reflection of C with respect to the point D. Prove that PE\perp CQ.
  • A circle passing through the midpoint M of the side BC and the vertex A of the triangle ABC intersects the segments AB and AC for the second time in the points P and Q, respectively. Prove that if \angle BAC=60^{\circ}, then AP+AQ+PQ<AB+AC+\frac{1}{2} BC.
  • Let P and Q be the midpoints of the sides BC and CD, respectively in a rectangle ABCD. Let K and M be the intersections of the line PD with the lines QB and QA, respectively, and let N be the intersection of the lines PA and QB. Let X, Y and Z be the midpoints of the segments AN, KN and AM, respectively. Let \ell_1 be the line passing through X and perpendicular to MK, \ell_2 be the line passing through Y and perpendicular to AM and \ell_3 the line passing through Z and perpendicular to KN. Prove that the lines \ell_1, \ell_2 and \ell_3 are concurrent.
  • Let ABC be an equilateral triangle , and P be a point on the circumcircle of the triangle but distinct from A ,B and C. The lines through P and parallel to BC , CA , AB intersect the lines CA , AB , BC at M , N and Q respectively .Prove that M , N and Q are collinear .
  • Let ABC be an isosceles triangle with AB=AC . Let also \omega be a circle of center K tangent to the line AC at C which intersects the segment BC again at H . Prove that HK \bot AB.
  • Let the circles k_1 and k_2 intersect at two points A and B, and let t be a common tangent of k_1 and k_2 that touches k_1 and k_2 at M and N respectively. If t\perp AM and MN=2AM, evaluate the angle NMB.
  • Let MNPQ be a square of side length 1 , and A , B , C , D points on the sides MN , NP , PQ and QM respectively such that AC \cdot BD=\frac{5}{4}. Can the set \{AB , BC , CD , DA \} be partitioned into two subsets S_1 and S_2 of two elements each , so that each one has the sum of his elements a positive integer?
  • Consider a triangle ABC with \angle ACB=90^{\circ}. Let F be the foot of the altitude from
    C. Circle \omega touches the line segment FB at point P, the altitude CF at point Q and the
    circumcircle of ABC at point R. Prove that points A, Q, R are collinear and AP = AC.

Problem Sets

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Problem Set 1
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Problem Set 2
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Problem Set 3
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Problem Set 4
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Problem Set 5
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Problem Set 6
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Problem Set 7
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Problem Set 8
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Problem Set 9
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Problem Set 10
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Lecture 1
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Lecture 2
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Lecture 3
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Lecture 4
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Lecture 5
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Lecture 6
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Lecture 7
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Lecture 8
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Lecture 9
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Lecture 10
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Lecture 11
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Lecture 12
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Lecture 13
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Lecture 14
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Lecture 15
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Lecture 16
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Lecture 17
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Lecture 18
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Lecture 19
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Lecture 20
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Lecture 21
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Lecture 22
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Lecture 23
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Lecture 24
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Lecture 25
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Lecture 26
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Lecture 27
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Lecture 28
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Lecture 29
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Lecture 30

Multiple-Choice Test

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Mock Test 1
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Mock Test 2
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Mock Test 3
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Mock Test 4
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Mock Test 5

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