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.

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

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