mechmat competition1998

Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University See William Lowell Putnam Math Competition, 1996, B1 See William Lowell Putnam Math Competition, 1989, A4 See William Lowell Putnam Math Competition, 1997, B6 Let q ∈ C, q = Prove that for every non-singular matrix A ∈ M atn×n (C) there exists non-singular matrix B ∈ M atn×n (C) such that AB − qBA = I (V Mazorchuk) See William Lowell Putnam Math Competition, 1992, B6 See William Lowell Putnam Math Competition, 1989, A6 See William Lowell Putnam Math Competition, 1997, B2 Does there exist a function f ∈ C(R) such that for every real number x we have f (x + t)dt = arctan x? See William Lowell Putnam Math Competition, 1997, A4 10 The sequence {xn , n ≥ 1} ⊂ R is defined as follows: √ x1 = 1, xn+1 = + { n}, n ≥ 1, + xn where {a} denotes fractional part of a Find the limit N x2k lim N →∞ N k=1 (A Kukush) (A Dorogovtsev, Jr.) 11 See William Lowell Putnam Math Competition, 1995, A5 12 Let B be complex Banach space and linear operators A, C ∈ L(B) be such that σ(AC ) {x + iy|x + y = 1} = ∅ Prove that σ(CAC) {x + iy|x + y = 1} = ∅ (A Dorogovtsev)