Řešitelský seminář, 14.3.2017
Problem 1. Let A be a unitary and commutative ring with an odd number of elements. If n is the number of solutions of the equation x2 = x, x € A, and m the number of invertible elemets, show that n divides m.
Problem 2. Let f : [0,1] —> R be a continuous differentiable function, such that
f {f'{x)fdx < 2 / f(x)dx. Jo Jo
Findfiff(l) = -\.
Problem 3. Prove or give an counterexample: Every connected, locally pathwise connected set in R™ is pathwise connected.
Domácí úloha
Problem 4. Show that a positive constant t can satisfy ex > xl for all x > 0, iff t < e.