NL-NWO 1990

Prove that for every integer $n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n-1)<n^n.$

Consider the sequence $a_1=\frac{3}{2}, a_{n+1}=\frac{3a_n^2+4a_n-3}{4a_n^2}.$ $(a)$ Prove that $1<a_n$ and $a_{n+1}<a_n$ for all $n$ . $(b)$ From $(a)$ it follows that $\lim_{n\to\infty}a_n$ exists. Find this limit. $(c)$ Determine $\lim_{n\to\infty}a_1a_2a_3...a_n$ .

A polynomial $f(x)=ax^4+bx^3+cx^2+dx$ with $a,b,c,d>0$ is such that $f(x)$ is an integer for $x \in \{ -2,-1,0,1,2 \}$ and $f(1)=1$ and $f(5)=70$ . $(a)$ Show that $a=\frac{1}{24}, b=\frac{1}{4},c=\frac{11}{24},d=\frac{1}{4}$ . $(b)$ Prove that $f(x)$ is an integer for all $x \in \mathbb{Z}$ .

If $ABCDEFG$ is a regular $7$ -gon with side $1$ , show that: $\frac{1}{AC}+\frac{1}{AD}=1$ .