Řešitelský seminář, 7. 3. 2017
Problem 1. Let f : R —> R be continuous. Suppose that R contains a countably infinite subset S such that
if p and q are not in S. Prove that f is identicaly zero.
Problem 2. Let n be a positive integer and A € A4„(C), A = (apq),l < p,q < n be such that ciij + cijk + ciki = 0 for all i, j, k € {1, 2 ... n}. Prove that rank(A) < 2.
Problem 3. Let (A, +, •) be a finite unitary and commutative ring. Denote by d the number of divisors of zero and by n the number of nilpotents. Prove
1. If x and y are nilpotents, then x + y and xy are nilpotents,
2. n is a divisor of d.
Problem 4. Prove or give an counterexample: Every connected, locally pathwise connected set in R™ is pathwise connected.
p
f (x) dx = 0
Domácí úloha
Problem 5. For which numbers a € (1, oo) is true that xa < ax for all x € (1, oo)?