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
2024 UGB

8 maths questions

Q1 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Find, with proof, all possible values of $t$ such that
$$\lim _ { n \rightarrow \infty } \left\{ \frac { 1 + 2 ^ { 1/3 } + 3 ^ { 1/3 } + \cdots + n ^ { 1/3 } } { n ^ { t } } \right\} = c$$
for some real number $c > 0$. Also find the corresponding values of $c$.
Q2 Proof Characterization or Determination of a Set or Class View
Suppose $n \geq 2$. Consider the polynomial
$$Q _ { n } ( x ) = 1 - x ^ { n } - ( 1 - x ) ^ { n } .$$
Show that the equation $Q _ { n } ( x ) = 0$ has only two real roots, namely 0 and 1.
Q3 Proof Direct Proof of a Stated Identity or Equality View
Let $ABCD$ be a quadrilateral with all internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta _ { 1 } , \Delta _ { 2 } , \Delta _ { 3 }$ and $\Delta _ { 4 }$ denote the areas of the shaded triangles shown. Prove that
$$\Delta _ { 1 } - \Delta _ { 2 } + \Delta _ { 3 } - \Delta _ { 4 } = 0 .$$
Q4 Proof Direct Proof of a Stated Identity or Equality View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function which is differentiable at 0. Define another function $g : \mathbb { R } \rightarrow \mathbb { R }$ as follows:
$$g ( x ) = \begin{cases} f ( x ) \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{cases}$$
Suppose that $g$ is also differentiable at 0. Prove that
$$g ^ { \prime } ( 0 ) = f ^ { \prime } ( 0 ) = f ( 0 ) = g ( 0 ) = 0$$
Q5 Proof Computation of a Limit, Value, or Explicit Formula View
Let $P ( x )$ be a polynomial with real coefficients. Let $\alpha _ { 1 } , \ldots , \alpha _ { k }$ be the distinct real roots of $P ( x ) = 0$. If $P ^ { \prime }$ is the derivative of $P$, show that for each $i = 1,2 , \ldots , k$,
$$\lim _ { x \rightarrow \alpha _ { i } } \frac { \left( x - \alpha _ { i } \right) P ^ { \prime } ( x ) } { P ( x ) } = r _ { i } ,$$
for some positive integer $r _ { i }$.
Q6 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
Let $x _ { 1 } , \ldots , x _ { 2024 }$ be non-negative real numbers with $\sum _ { i = 1 } ^ { 2024 } x _ { i } = 1$. Find, with proof, the minimum and maximum possible values of the expression
$$\sum _ { i = 1 } ^ { 1012 } x _ { i } + \sum _ { i = 1013 } ^ { 2024 } x _ { i } ^ { 2 }$$
Q7 Connected Rates of Change Optimizing a Rate of Change Over Time View
Consider a container of the shape obtained by revolving a segment of the parabola $x = 1 + y ^ { 2 }$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ into the container. Let $h ( t )$ be the height of water inside the container at time $t$. Find the time $t$ when the rate of change of $h ( t )$ is maximum.
Q8 Proof Direct Proof of an Inequality View
In a sports tournament involving $N$ teams, each team plays every other team exactly once. At the end of every match, the winning team gets 1 point and the losing team gets 0 points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows:
$$x _ { 1 } \geq x _ { 2 } \geq \cdots \geq x _ { N }$$
Prove that for any $1 \leq k \leq N$,
$$\frac { N - k } { 2 } \leq x _ { k } \leq N - \frac { k + 1 } { 2 } .$$