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 nth 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


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.

Solution: Click Here

To check all the previous weekly challenging problems and discussion: Click here

We will post the solutuions next week along with a new set of problem.

Leave a Reply