Week 4 Challenging Problems

Fractions > Week 4 Challenging Problems

Problems for the week

Deadline 25.05.2020

Problem 1.

Prove that when $n$ is a sufficiently large positive integer there exists a finite set $S$ of prime numbers such that the sum of $[\frac{n}{p}]$ over $p\in S$ is equal to $n$ .

Solution: Click Here

Problem 2.

Let $H_n$ denote the $n$ th harmonic number $\sum^n_1 \frac{1}{k}$ . Let $H_0=0$ .Prove that for positive integers $n$ and $k$ with $k \leq n$ ,

$\sum^{k-1}_{i=0}\sum^n_{j=k}(-1)^{i+j-1}\binom{n}{i}\binom{n}{j}\frac{1}{j-i}=\sum^{k-1}_{i=0} \binom{n}{i}^2 (H_{n-i}-H_i).$

Solution: Click Here

Problem 3.

Find all pairs $(x,y)$ of integers such that

$x^2+3xy+4006(x+y)+2003^2=0$

Solution: Click Here

Problem 4.

Let $f:\mathbb{R}\to\mathbb{R}$ be diffrentiable, with $f(0)=1,$ and with $f$ and $f^'$ both nowhere zero on $\mathbb{R}$ . Let $a_1$ be a positive real number, and for $n\geq 1$ let $a_{n+1}=a_n f(a_n)$ . Prove that $\sum^{\infty}_{n=1} a_n$ is divergent.