isi-entrance

Papers (32)
2026
UGA 30 UGB 8
2024
UGA 30 UGB 8
2023
UGA 30 UGB 8
2022
UGA 30 UGB 9
2021
UGA 30 UGB 8
2020
UGA 30 UGB 8
2019
UGA 30 UGB 8
2018
UGA 30 UGB 8
2017
UGA 30 UGB 8
2016
UGA 73 UGB 73
2015
UGA 39 UGB 39
2014
solved 22
2013
isi-entrance_2013.pdf 73
2012
solved 29
2011
solved 19
2010
solved 21
2009
solved 10
2007
solved 6
2006
solved 9
2005
solved 10
Unknown
subjective_collection 10
2021 UGB

8 maths questions

Q1 Proof Existence Proof View
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).
Q2 Proof Characterization or Determination of a Set or Class View
Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below.
$$\begin{aligned} f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\ f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y ) \end{aligned}$$
for all integers $x , y$.
(i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.
Q3 Number Theory Combinatorial Number Theory and Counting View
Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.
Q4 Applied differentiation Properties of differentiable functions (abstract/theoretical) View
Let $g : ( 0 , \infty ) \rightarrow ( 0 , \infty )$ be a differentiable function whose derivative is continuous, and such that $g ( g ( x ) ) = x$ for all $x > 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.
Q5 Roots of polynomials Polynomial evaluation, interpolation, and remainder View
Let $a _ { 0 } , a _ { 1 } , \cdots , a _ { 19 } \in \mathbb { R }$ and
$$P ( x ) = x ^ { 20 } + \sum _ { i = 0 } ^ { 19 } a _ { i } x ^ { i } , \quad x \in \mathbb { R }$$
If $P ( x ) = P ( - x )$ for all $x \in \mathbb { R }$, and
$$P ( k ) = k ^ { 2 } , \text{ for } k = 0,1,2 \cdots , 9$$
then find
$$\lim _ { x \rightarrow 0 } \frac { P ( x ) } { \sin ^ { 2 } x }$$
Q6 Proof Existence Proof View
If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.
Q7 Proof Direct Proof of an Inequality View
Let $a , b , c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a + b + c = 6$ and $a b + b c + a c = 9$. Suppose $a < b < c$. Show that
$$0 < a < 1 < b < 3 < c < 4$$
Q8 Connected Rates of Change Volume/Height Related Rates for Containers and Solids View
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base. The depth of the pond is 6 m. The square at the bottom has side length 2 m and the top square has side length 8 m. Water is filled in at a rate of $\frac { 19 } { 3 }$ cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?