Week 2 Challenging Problems

Fractions > Week 2 Challenging Problems

Problems for the week

Determine all positive integers $K$ such that,

$\frac{\tau{(n^2)}}{\tau{(n)}}=k$

for some $n$ , where $\tau(n)$ is the number of divisors of $n$ .

Find the first digit before and after the decimal point in $\bigg(\sqrt{2}+\sqrt{3}\bigg)^{1980}$ . (without using a calculator)

$2n$ points are chosen on a circle. In how many ways can you join pairs of points by nonintersecting chords?

Let $f:[a,b] \to \mathbb{R}$ be a function, continuous on [a,b] and differentiable on (a,b). Prove that if there exists $c\in (a,b)$ such that

$\frac{f(b)-f(c)}{f(c)-f(a)}<0$

then there exists $\xi \in (a,b)$ such that $f'(\xi)=0.$