Řešitelský seminář, 18.4.2017
Problem 1. Let p(x) be the polynomial x3 + 14a;2 — 2x + 1 Let p(n\x) denotes p(p(n ^(x)). Show that there is an integer N such that p(N^(x) — x is divisible by 101 for all integers x.
Problem 2. Find the number of integers c such that —2007 < c < 2007 and there exists an integer x such that x2 + c is a multiple of 22007.
Domácí úloha
Problem 3. Let (a„) be a sequence of nonzero real numbers. Prove that the sequence of functions fn : R —> R
fn{x) = — sin(a„a;) + cos(a; + a„) has a subsequence converging to a continuous function.