Problem 1
Evaluate $\displaystyle \int_1^4 \frac{x-2}{(x^2 + 4) \sqrt{x}} dx$.
Problem 2
A sequence $(a_n)$ is defined by $a_0 = -1$, $a_1 = 0$, and $$ a_{n+1} = a_n^2 - (n+1)^2 a_{n-1} - 1 $$ for all positive integers $n$. Find $a_{100}$.
Problem 3
Find $\displaystyle \sum_{k=1}^\infty \frac{k^2 - 2}{(k + 2)!}$.
Problem 4
Let $m, n$ be positive integers and let $[a]$ denote the residue class $\bmod\ mn$ of the integer $a$ (thus $\lbrace [r] \mid r \text{ is an integer} \rbrace$ has exactly $mn$ elements). Suppose the set $\lbrace [ar] \mid r \text{ is an integer} \rbrace$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is prime to $mn$ and $[nq] = [a]$.
Problem 5
Find $\displaystyle \lim_{x \to \infty} (2x)^{1+\frac{1}{2x}} - x^{1+\frac{1}{x}} - x$.
Problem 6
Let $S$ be a set with an asymmetric relation $<$; this means that if $a, b \in S$ and $a < b$, then we do not have $b < a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a, b \in S$, then $a < b$ if and only if $a \prec b$, and if $x, y \in T$ with $x \prec y$, then there exists $t \in T$ such that $x \prec t \prec y$ ($t \in T$ means “$t$ is an element of $T$”).
Problem 7
Let $P(x) = x^{100} + 20x^{99} + 198x^{98} + a_{97} x^{97} + \dots + a_1 x + 1$ be a polynomial where the $a_i$ ($1 \leq i \leq 97$) are real numbers. Prove that the equation $P(x) = 0$ has at least one complex root (i.e., a root of the form $a + bi$ with $a, b$ real numbers and $b \ne 0$).