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
2005 solved

10 maths questions

Q1 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
In a right angle triangle with sides $a < b < c$, where $\angle ACB = \theta$ is the smallest angle, show that $\sin^2\theta - \sqrt{5}\sin\theta + 1 = 0$, given that $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$ (i.e., the reciprocals of the sides also form a right triangle).
Q2 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.
Q3 Sequences and series, recurrence and convergence Summation of sequence terms View
Let $f(a, b)$ be a function satisfying $f(a, b) = f(a, c) + f(c, b) - 2f(a,c)f(c,b)$ with $f(99, 100) = 1/3$. Find $f(1, 100)$.
Q4 Trig Proofs Trigonometric Inequality Proof View
Show that $\sin^5 x + \cos^3 x \geq \sin^3 x + \cos^2 x$ implies the expression equals $1$, and find when equality holds.
Q5 Sine and Cosine Rules Prove an inequality or ordering relationship in a triangle View
In a triangle with angles $P$, $Q$, $R$, let $\alpha$, $\beta$, $\gamma$ be the angles $\angle QCR = 2P$, $\angle QIR = Q + R$, $\angle QOR = P + Q/2 + R/2$ respectively. Show that $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} > \frac{1}{45}$.
Q6 Curve Sketching Sketching a Curve from Analytical Properties View
Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.
Q7 Combinations & Selection Combinatorial Identity or Bijection Proof View
Let $A_{m,n}$ denote the set of strictly increasing sequences $1 \leq \alpha_1 < \alpha_2 < \cdots < \alpha_m \leq n$ of integers, $B_{m,n}$ denote the set of non-negative integer solutions of $\alpha_1 + \alpha_2 + \cdots + \alpha_m = n$, and $C_{m,n}$ denote the set of strictly increasing sequences chosen from $\{1,2,\ldots,n\}$.
a) Construct a bijection from $A_{m,n}$ to $B_{m+1,n-1}$.
b) Construct a bijection from $A_{m,n}$ to $C_{m,m+n-1}$.
c) Find the number of elements in $A_{m,n}$.
Q8 Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $g(n) = 5^k$ where $k$ is the number of distinct primes dividing $n$, and let $h(n) = 0$ if $n$ is divisible by $k^2$ for some integer $k > 1$, and $h(n) = 1$ otherwise.
a) Show that $g(mn) = g(m)g(n)$ does not hold in general, and determine when it holds.
b) Show that $h(mn) = h(m)h(n)$ for all positive integers $m, n$.
Q9 Number Theory Combinatorial Number Theory and Counting View
a) Show that among any 2-coloring (Red/Blue) of the vertices of a regular hexagon and its center, there must exist a monochromatic equilateral triangle.
b) Extend the argument to show the result holds for a larger configuration of points.
Q10 Number Theory Combinatorial Number Theory and Counting View
a) $n$ lines are drawn through a point $A$ inside a circle, creating chords. Show that the number of regions created inside the circle is $(n+1)^2$, assuming no three lines meet at any point other than $A$.
b) Using the result from part a), find the total number of regions when additional lines are drawn, and show the total is $(n+1)^2 + (2n+1)n$.