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
2007 solved

6 maths questions

Q1 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.
Q3 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Show that $\int_1^n [u]([u]+1)f(u)\,du = 2\sum_{i=1}^{[n]} i \int_i^{i+1} f(u)\,du$ (or an equivalent integral identity involving the floor function).
Q5 Harmonic Form View
Find the range of $y = \cos\theta\left(\sin\theta + \sqrt{\sin^2\theta + 3}\right)$.
Q7 Stationary points and optimisation Geometric or applied optimisation problem View
Let $a$, $b$, $h$ be the three edges meeting at a particular vertex of a triangular prism, such that $a$, $b$ are sides of a base triangle with angle $\theta$ between them and $h$ is the height of the prism. Given that the total surface area is $K$, show that the volume $V$ satisfies $V \leq \sqrt{K^3/54}$, and find the dimensions of the prism of maximum volume.
Q8 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
In how many ways can the numbers $1, 2, \ldots, 9$ be arranged in a $3 \times 3$ grid \[ \begin{array}{|c|c|c|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{array} \] such that each row and each column is in increasing order (i.e., $A < B < C$, $D < E < F$, $G < H < I$, $A < D < G$, $B < E < H$, $C < F < I$)?
Q10 Number Theory Linear Diophantine Equations View
Let $n_1, n_2, \ldots, n_k$ be positive integers with $\gcd(n_1, n_2, \ldots, n_k) = 1$. Show that every sufficiently large positive integer $n$ can be represented as $n = \sum_i c_i n_i$ with $c_i \geq 0$ integers.