**Problem 1.**

A Dyck -path is a lattice path of upsteps (1,1) and downsteps (1,-1) that starts at the origin and never falls below the -axis. Show that the number of Dyck -paths that avoid is twice the number of Dyck – paths.

A Dyck -path is a lattice path of upsteps (1,1) and downsteps (1,-1) that starts at the origin and never falls below the -axis. Show that the number of Dyck -paths that avoid is twice the number of Dyck – paths.

Given a positive integer , find the minimum value of

subject to the condition that be distinct positive integers.

Let be a monic polynomial with complex coefficients and with zeros . Prove that

For complex , let be a map of onto .Let be the pseudohyperbolic distance.

(a) Prove that there exists a function so that

for every .

(b) Find the minimal value of for which bound holds.

I guess problem 2 ans is n(n+1)/2 …

Solutions have been posted along with the Week 6 challenging problems. Check them out.