Řešitelský seminář, 21.2.2017
Problem 1. Let A and B be 2 x 2 real matrices such that AB = A2B2 - (AB)2 and det(B) = 2. Evaluate det(A + 2B) - det(B + 2A).
Problem 2. Suppose that / : R -> 1 is a nonconstant function such that f(x) < f(y) whenever x < y. Prove that there exist a£l and c > 0 such that /(a + x) — /(a — x) > cx for all x € [0,1].
3
Problem 3. Prove that for all x > 0, sin a; >£ — %-.
7 6
Problem 4. Which number is larger, it3 or 371" ?
Problem 5. Let A be a linear transformation on R3 whose matrix (relative to the usual basis for W3) is both symmetric and orthogonal. Prove that A is either plus or minus the identity, or a rotation by 18CP about some axis, or a reflection about some two-dimensional subspace of R3.
Problem 6. Let n > 2 be an integer and let (K, +, •) be a commutative Geld with the property:
1 + • - + 1 ^ 0, m = 2,..., n.
m times
Consider a polynomial f € K[x] of degree n and G a subgroup of the aditive group (K, +), G ^ K. Prove that there exists a € K, such that /(a) ^ G.