## IMC 2016 Problems – Day 2

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

**Problem 7.** Today, Ivan the Confessor prefers continuous functions satisfying for all . Fin the minimum of over all preferred functions.

**Problem 8.** Let be a positive integer and denote by the ring of integers modulo . Suppose that there exists a function satisfying the following three properties:

- (i) ,
- (ii) ,
- (iii) for all .

Prove that modulo .

**Problem 9.** Let be a positive integer. For each nonnegative integer let be the number of solutions of the inequality . Prove that for every we have .

**Problem 10.** Let be a complex matrix whose eigenvalues have absolute value at most . Prove that

(Here for every matrix and for every complex vector .)

Official source and more infos here.

Advertisements

Categories: Olympiad, Uncategorized
IMC, olympiad, problems

Comments (0)
Trackbacks (0)
Leave a comment
Trackback