Problem 1
Evaluate $\displaystyle{\int_{0}^{x} \frac{d\theta}{2 + \tan \theta}},$ where $0 \le x \le \pi/2$.
Use your result to show that $\displaystyle{\int_{0}^{\pi/4} \frac{d\theta}{2 + \tan \theta} = \frac{\pi + \ln(9/8)}{10}}$.
Problem 2
Given that $e^x = 1/0! + x/1! + x^2/2! + \dots + x^n/n! + \dots$ find, in terms of $e$, the exact values of
- $\dfrac{1}{1!} + \dfrac{2}{3!} + \dfrac{3}{5!} + \dots + \dfrac{n}{(2n - 1)!} + \dots$ and
- $\dfrac{1}{3!} + \dfrac{2}{5!} + \dfrac{3}{7!} + \dots + \dfrac{n}{(2n + 1)!} + \dots$
Problem 3
Solve the initial value problem $\dfrac{dy}{dx} = y \ln y + y e^x$, $y(0) = 1$ (i.e., find $y$ in terms of $x$).
Problem 4
In the diagram below, $P, Q, R$ are points on $BC, CA, AB$ respectively such that the lines $AP, BQ, CR$ are concurrent at $X$. Also $PR$ bisects $\angle BRC$, i.e. $\angle BRP = \angle PRC$. Prove that $\angle PRQ = 90^{\circ}$.
Problem 5
Find the third digit after the decimal point of $$ (2 + \sqrt{5})^{100} ((1 + \sqrt{2})^{100} + (1 + \sqrt{2})^{-100}). $$ For example, the third digit after the decimal point of $\pi = 3.14159 \dots$ is $1$.
Problem 6
Let $n$ be a positive integer, let $A, B$ be square symmetric $n \times n$ matrices with real entries (so if $a_{ij}$ are the entries of $A$, the $a_{ij}$ are real numbers and $a_{ij} = a_{ji}$). Suppose there are $n \times n$ matrices $X, Y$ (with complex entries) such that $\det(AX + BY) \ne 0$. Prove that $\det(A^2 + B^2) \ne 0$ ($\det$ indicates the determinant).
Problem 7
Determine whether the series $\displaystyle \sum_{n=2}^{\infty} n^{-(1+(\ln(\ln n))^{-2})}$ is convergent or divergent ($\ln$ denotes natural log).