VTRMC 2011

Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$

A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$ , and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$ . Find $a_{100}$ .

Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$ .

Let $m,n$ be positive integers and let $[a]$ denote the residue class $\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$ .

Find $\lim_{x\to\infty}\left((2x)^{1+\frac1{2x}}-x^{1+\frac1x}-x\right)$ .

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 $x\prec t\prec y$ .

Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.