Olympiad problems

2020 ISI Entrance Examination, 4

Let a real-valued sequence $\{x_n\}_{n\geqslant 1}$ be such that $$\lim_{n\to\infty}nx_n=0$$ Find all possible real values of $t$ such that $\lim_{n\to\infty}x_n\big(\log n\big)^t=0$ .

2020 ISI Entrance Examination, 1

Tags: isi , 2020 , polynomial , complex numbers , algebra

Let $i$ be a root of the equation $x^2+1=0$ and let $\omega$ be a root of the equation $x^2+x+1=0$ . Construct a polynomial $$f(x)=a_0+a_1x+\cdots+a_nx^n$$ where $a_0,a_1,\cdots,a_n$ are all integers such that $f(i+\omega)=0$ .

2021 ISI Entrance Examination, 1

Tags: combinatorics , isi

There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $(n+1)$ friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

2019 ISI Entrance Examination, 5

Tags: isi , Indian Statistical Institute , 2019 , convex quadrilateral , Convex Set , geometry

A subset $\bf{S}$ of the plane is called [i]convex[/i] if given any two points $x$ and $y$ in $\bf{S}$, the line segment joining $x$ and $y$ is contained in $\bf{S}$. A quadrilateral is called [i]convex[/i] if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area $1$, there is a rectangle $R$ of area $2$ such that $Q$ can be drawn inside $R$.

2021 ISI Entrance Examination, 3

Tags: number theory , algebra , power series , isi

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

2022 ISI Entrance Examination, 9

Tags: isi , Indian Statistical Institute , complex numbers , 2022 , algebra

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n=2$. Then show that the same $k$ works for any $n \geqslant 2$.]

2022 ISI Entrance Examination, 3

Tags: isi , Indian Statistical Institute , coordinate geometry , 2022

Consider the parabola $C: y^{2}=4 x$ and the straight line $L: y=x+2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_{1}$ and $Q_{2}$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_{1}$ and $Q_{2}$. Find the locus of $Q$ as $P$ moves along $L$.

2020 ISI Entrance Examination, 7

Tags: isi , 2020 , number theory

Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then [b](a)[/b] Prove that $d\leqslant 2$. [b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .

2022 ISI Entrance Examination, 6

Tags: isi , Indian Statistical Institute , 2022 , Sequence , limit , real analysis

Consider a sequence $P_{1}, P_{2}, \ldots$ of points in the plane such that $P_{1}, P_{2}, P_{3}$ are non-collinear and for every $n \geq 4, P_{n}$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_{1}$ and $P_{5}$. Prove the following: [list=a] [*] The area of the triangle formed by the points $P_{n}, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity. [*] The point $P_{9}$ lies on $L$. [/list]

2022 ISI Entrance Examination, 5

Tags: isi , Indian Statistical Institute , number theory , 2022

For any positive integer $n$, and $i=1,2$, let $f_{i}(n)$ denote the number of divisors of $n$ of the form $3 k+i$ (including $1$ and $n$ ). Define, for any positive integer $n$, $$f(n)=f_{1}(n)-f_{2}(n)$$ Find the value of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.

2022 ISI Entrance Examination, 1

Tags: isi , Indian Statistical Institute , 2022 , combinatorics

Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by $0$ or $1$ . [list=a] [*] In how many ways can this be done such that each row sum and each column sum is even? [*] In how many ways can this be done such that each row sum and each column sum is odd? [/list]

2015 ISI Entrance Examination, 5

Tags: algebra , isi

If $0<a_1< \cdots < a_n$, show that the following equation has exactly $n$ roots. $$ \frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+ \frac{a_3}{a_3-x}+ \cdots + \frac {a_n}{a_n - x} = 2015$$

2020 ISI Entrance Examination, 2

Tags: isi , 2020

Let $a$ be a fixed real number. Consider the equation $$ (x+2)^{2}(x+7)^{2}+a=0, x \in R $$ where $R$ is the set of real numbers. For what values of $a$, will the equ have exactly one double-root?

2018 ISI Entrance Examination, 4

Tags: isi , 2018 , Indian Statistical Institute , calculus

Let $f:(0,\infty)\to\mathbb{R}$ be a continuous function such that for all $x\in(0,\infty)$, $$f(2x)=f(x)$$ Show that the function $g$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$ is a constant function.

2019 ISI Entrance Examination, 2

Tags: isi , Indian Statistical Institute , 2019 , calculus

Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$ [b](a)[/b] Show that $f$ has exactly one point of discontinuity. [b](b)[/b] Evaluate $f$ at its point of discontinuity.

2022 ISI Entrance Examination, 7

Tags: isi , Indian Statistical Institute , 2022 , polynomial , limit

Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$ Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$

2018 ISI Entrance Examination, 1

Tags: isi , 2018 , Indian Statistical Institute

Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$

2007 ISI B.Math Entrance Exam, 6

Tags: isi , ISI entrance , combinatorics

In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have????

2023 ISI Entrance UGB, 7

Tags: algebra , polynomial , isi , Indian Statistical Institute

(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$. (b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.

2019 ISI Entrance Examination, 8

Tags: isi , Indian Statistical Institute , 2019 , distance , coordinate geometry , real analysis

Consider the following subsets of the plane:$$C_1=\Big\{(x,y)~:~x>0~,~y=\frac1x\Big\} $$ and $$C_2=\Big\{(x,y)~:~x<0~,~y=-1+\frac1x\Big\}$$ Given any two points $P=(x,y)$ and $Q=(u,v)$ of the plane, their distance $d(P,Q)$ is defined by $$d(P,Q)=\sqrt{(x-u)^2+(y-v)^2}$$ Show that there exists a unique choice of points $P_0\in C_1$ and $Q_0\in C_2$ such that $$d(P_0,Q_0)\leqslant d(P,Q)\quad\forall ~P\in C_1~\text{and}~Q\in C_2.$$

2022 ISI Entrance Examination, 4

Tags: isi , Indian Statistical Institute , polynomial

Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show that the equation $P(P(x))=0$ has at least as many distinct real roots as the equation $P(x)=0$.

2021 ISI Entrance Examination, 2

Tags: functional equation , isi , ISI entrance

Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers $x,y$. [b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$. [b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.

2018 ISI Entrance Examination, 3

Tags: isi , Indian Statistical Institute , function , calculus , continuous

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$, $$f(x)=f(e^tx)$$ Show that $f$ is a constant function.

2019 ISI Entrance Examination, 7

Tags: isi , Indian Statistical Institute , 2019 , polynomial , Integer Polynomial , algebra

Let $f$ be a polynomial with integer coefficients. Define $$a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$$ and $~a_n = f(a_{n-1})$ for $n \geqslant 3$. If there exists a natural number $k \geqslant 3$ such that $a_k = 0$, then prove that either $a_1=0$ or $a_2=0$.

2018 ISI Entrance Examination, 5

Tags: isi , 2018 , Indian Statistical Institute , calculus

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$