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
2026 UGB

8 maths questions

Q1 10 marks Proof Existence Proof View
Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is differentiable and $\left| f ^ { \prime } ( x ) \right| < \frac { 1 } { 2 }$ for all $x \in \mathbb { R }$. Show that for some $x _ { 0 } \in \mathbb { R } , f \left( x _ { 0 } \right) = x _ { 0 }$.
Q2 10 marks Trig Proofs Trigonometric Equation Constraint Deduction View
If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C = 2 \left( \cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C \right) ,$$ prove that the triangle must have a right angle.
Q3 10 marks Proof Direct Proof of a Stated Identity or Equality View
Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.
Q4 10 marks Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $S ^ { 1 } = \{ z \in \mathbb { C } | | z \mid = 1 \}$ be the unit circle in the complex plane. Let $f : S ^ { 1 } \rightarrow S ^ { 1 }$ be the map given by $f ( z ) = z ^ { 2 }$. We define $f ^ { ( 1 ) } : = f$ and $f ^ { ( k + 1 ) } : = f \circ f ^ { ( k ) }$ for $k \geq 1$. The smallest positive integer $n$ such that $f ^ { ( n ) } ( z ) = z$ is called the period of $z$. Determine the total number of points in $S ^ { 1 }$ of period 2025. (Hint: $2025 = 3 ^ { 4 } \times 5 ^ { 2 }$)
Q5 10 marks Proof Direct Proof of a Stated Identity or Equality View
Let $a , b , c$ be nonzero real numbers such that $a + b + c \neq 0$. Assume that $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = \frac { 1 } { a + b + c }$$ Show that for any odd integer $k$, $$\frac { 1 } { a ^ { k } } + \frac { 1 } { b ^ { k } } + \frac { 1 } { c ^ { k } } = \frac { 1 } { a ^ { k } + b ^ { k } + c ^ { k } }$$
Q6 10 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
Let $\mathbb { N }$ denote the set of natural numbers, and let $\left( a _ { i } , b _ { i } \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb { N } \times \mathbb { N }$. Show that there are three distinct elements in the set $\left\{ 2 ^ { a _ { i } } 3 ^ { b _ { i } } : 1 \leq i \leq 9 \right\}$ whose product is a perfect cube.
Q7 10 marks Proof Existence Proof View
Consider a ball that moves inside an acute-angled triangle along a straight line, until it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence $=$ angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.
Q8 10 marks Proof Direct Proof of an Inequality View
Let $n \geq 2$ and let $a _ { 1 } \leq a _ { 2 } \leq \cdots \leq a _ { n }$ be positive integers such that $\sum _ { i = 1 } ^ { n } a _ { i } = \Pi _ { i = 1 } ^ { n } a _ { i }$. Prove that $\sum _ { i = 1 } ^ { n } a _ { i } \leq 2n$ and determine when equality holds.