Week 5 Challenging Problems

Fractions > Week 5 Challenging Problems

Problems for the week

Deadline 02.06.2020

Problem 1.

A Dyck $n$ -path is a lattice path of $n$ upsteps (1,1) and $n$ downsteps (1,-1) that starts at the origin and never falls below the $x$ -axis. Show that the number of Dyck $(2n)$ -paths that avoid $\{(4k,0):1\leq k\leq n-1\}$ is twice the number of Dyck $(2n-1)$ – paths.

Problem 2.

Given a positive integer $n$ , find the minimum value of

$\frac{x^3_1+\dots+x^3_n}{x_1+\dots+x_n}$

subject to the condition that $x_1,\dots,x_n$ be distinct positive integers.

Problem 3.

Let $A(z) = \sum^n_{k=0} a_k z^k$ be a monic polynomial with complex coefficients and with zeros $z_1,\dots , z_n$ . Prove that

$\frac{1}{n}\sum^n_{k=1}|z_k|^2<1+ \max_{1\leq k\leq n }|a_{n-k}|^2.$

Problem 4.

For complex $a,z \in \mathbb{D}={s:|s|<1}$ , let $F(a,z)=\frac{(a+z)}{(1+\bar{a}z)}$ be a map of $\mathbb{D}$ onto $\mathbb{D}$ .Let $\rho(a,b)=|\frac{(a-b)}{(1-\bar{a}b)}|$ be the pseudohyperbolic distance.
(a) Prove that there exists a function $C:\mathbb{D}\to\mathbb{R}^{+}$ so that

$\rho(F(a,z),F(b,z))\leq C(z)\rho(a,b)$

for every $a,b,z\in\mathbb{D}$ .
(b) Find the minimal value of $C(z)$ for which bound holds.

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.

2 Comments

Sounak Das

June 8, 2020 at 2:47 pm Reply

I guess problem 2 ans is n(n+1)/2 …
- FractionsHub
  
  July 4, 2020 at 1:42 am Reply
  
  Solutions have been posted along with the Week 6 challenging problems. Check them out.