# Problems for the week

#### 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.

#### Problem 2.

Given a positive integer , find the minimum value of

subject to the condition that be distinct positive integers.

#### Problem 3.

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

#### Problem 4.

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.