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
2023 UGB

8 maths questions

Q1 Number Theory Properties of Integer Sequences and Digit Analysis View
Determine all integers $n > 1$ such that every power of $n$ has an odd number of digits.
Q2 Addition & Double Angle Formulae Half-Angle Formula Evaluation View
Let $a _ { 0 } = \frac { 1 } { 2 }$ and $a _ { n }$ be defined inductively by
$$a _ { n } = \sqrt { \frac { 1 + a _ { n - 1 } } { 2 } } , n \geq 1 .$$
(a) Show that for $n = 0,1,2 , \ldots$,
$$a _ { n } = \cos \theta _ { n } \text { for some } 0 < \theta _ { n } < \frac { \pi } { 2 } ,$$
and determine $\theta _ { n }$.
(b) Using (a) or otherwise, calculate
$$\lim _ { n \rightarrow \infty } 4 ^ { n } \left( 1 - a _ { n } \right)$$
Q3 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
In a triangle $A B C$, consider points $D$ and $E$ on $A C$ and $A B$, respectively, and assume that they do not coincide with any of the vertices $A , B , C$. If the segments $B D$ and $C E$ intersect at $F$, consider the areas $w , x , y , z$ of the quadrilateral $A E F D$ and the triangles $B E F , B F C , C D F$, respectively.
(a) Prove that $y ^ { 2 } > x z$.
(b) Determine $w$ in terms of $x , y , z$.
Q4 Number Theory Divisibility and Divisor Analysis View
Let $n _ { 1 } , n _ { 2 } , \cdots , n _ { 51 }$ be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance, $2 ^ { 2022 }$ has exactly 2023 positive integer factors $1,2,2 ^ { 2 } , \cdots , 2 ^ { 2021 } , 2 ^ { 2022 }$. Assume that no prime larger than 11 divides any of the $n _ { i }$'s. Show that there must be some perfect cube among the $n _ { i }$'s. You may use the fact that $2023 = 7 \times 17 \times 17$.
Q5 Sequences and Series Recurrence Relations and Sequence Properties View
There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t _ { n }$ denote the number of ways this can be done. For example, clearly $t _ { 1 } = 2$ because we can have either a red or a blue tile. Also, $t _ { 2 } = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that $t _ { 2 n + 1 } = t _ { n } \left( t _ { n - 1 } + t _ { n + 1 } \right)$ for all $n > 1$.
(b) Prove that $t _ { n } = \sum _ { d \geq 0 } \binom { n - d } { d } 2 ^ { n - 2 d }$ for all $n > 0$.
Here,
$$\binom { m } { r } = \begin{cases} \frac { m ! } { r ! ( m - r ) ! } , & \text { if } 0 \leq r \leq m , \\ 0 , & \text { otherwise } , \end{cases}$$
for integers $m , r$.
Q6 Sequences and series, recurrence and convergence Proof by induction on sequence properties View
Let $\left\{ u _ { n } \right\} _ { n \geq 1 }$ be a sequence of real numbers defined as $u _ { 1 } = 1$ and
$$u _ { n + 1 } = u _ { n } + \frac { 1 } { u _ { n } } \text { for all } n \geq 1 .$$
Prove that $u _ { n } \leq \frac { 3 \sqrt { n } } { 2 }$ for all $n$.
Q7 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
(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 $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$.
(b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, 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 $n$.
Q8 Proof Deduction or Consequence from Prior Results View
Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function which is differentiable on $( 0,1 )$. Prove that either $f$ is a linear function $f ( x ) = a x + b$ or there exists $t \in ( 0,1 )$ such that $| f ( 1 ) - f ( 0 ) | < \left| f ^ { \prime } ( t ) \right|$.