Problem 1
Evaluate $\displaystyle{\int_{0}^{1} \int_{0}^{x} \int_{0}^{1-x^2} e^{(1-z)^2} dz dy dx}$.
Problem 2
Let $f$ be continuous real function, strictly increasing in an interval $[0, a]$ such that $f(0) = 0$. Let $g$ be the inverse of $f$, i.e., $g(f(x)) = x$ for all $x$ in $[0, a]$. Show that for $0 \le x \le a$, $0 \le y \le f(a)$, we have $$ xy \le \int_{0}^{x} f(t) dt + \int_{0}^{y} g(t) dt. $$
Problem 3
Find all continuously differentiable solutions $f(x)$ for $$ f(x)^2 = \int_{0}^{x} (f(t)^2 - f(t)^4 + (f'(t))^2) dt + 100 $$ where $f(0)^2 = 100$.
Problem 4
Consider the polynomial equation $ax^4 + bx^3 + x^2 + bx + a = 0$, where $a$ and $b$ are real numbers, and $a > 1/2$. Find the maximum possible value of $a + b$ for which there is at least one positive real root of the above equation.
Problem 5
Let $f : \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ be a function which satisfies $f(0, 0) = 1$ and $$ f(m, n) + f(m + 1, n) + f(m, n + 1) + f(m + 1, n + 1) = 0 $$ for all $m, n \in \mathbb{Z}$ (where $\mathbb{Z}$ and $\mathbb{R}$ denote the set of all integers and all real numbers, respectively). Prove that $|f(m, n)| \ge 1/3$, for infinitely many pairs of integers $(m, n)$.
Problem 6
Let $A$ be an $n \times n$ matrix and let $\alpha$ be an $n$-dimensional vector such that $A\alpha = \alpha$. Suppose that all the entries of $A$ and $\alpha$ are positive real numbers. Prove that $\alpha$ is the only linearly independent eigenvector of $A$ corresponding to the eigenvalue 1. Hint: if $\beta$ is another eigenvector, consider the minimum of $\alpha_i /|\beta_i|$, $i = 1, \dots, n$, where the $\alpha_i$’s and $\beta_i$’s are the components of $\alpha$ and $\beta$, respectively.
Problem 7
Define $f(1) = 1$ and $f(n + 1) = 2\sqrt{f(n)^2 + n}$ for $n \ge 1$. If $N \ge 1$ is an integer, find $\displaystyle{\sum_{n=1}^{N} \frac{1}{f(n)^2}}$.
Problem 8
Let a sequence $\lbrace x_n \rbrace_{n=0}^{\infty}$ of rational numbers be defined by $x_0 = 10$, $x_1 = 29$ and $x_{n+2} = \frac{19x_{n+1}}{94x_n}$ for $n \ge 0$. Find $\displaystyle{\sum_{n=0}^{\infty} \frac{x_{6n}}{2^n}}$.