mechmat competition1998

Competition for University Students

of Mechanics and Mathematics Faculty
of Kyiv State Taras Shevchenko University.
1. See William Lowell Putnam Math. Competition, 1996, B1.
2. See William Lowell Putnam Math. Competition, 1989, A4.
3. See William Lowell Putnam Math. Competition, 1997, B6.
4. Let q ∈ C, q = 1. 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)
5. See William Lowell Putnam Math. Competition, 1992, B6.
6. See William Lowell Putnam Math. Competition, 1989, A6.
7. See William Lowell Putnam Math. Competition, 1997, B2.
8. Does there exist a function f ∈ C(R) such that for every real number x we have
1

f (x + t)dt = arctan x?
0

9. See William Lowell Putnam Math. Competition, 1997, A4.
10. The sequence {xn , n ≥ 1} ⊂ R is defined as follows:
√
1
x1 = 1, xn+1 =
+ { n}, n ≥ 1,
2 + xn
where {a} denotes fractional part of a. Find the limit
N
1
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 2 ) {x + iy|x + y = 1} = ∅.
Prove that
σ(CAC) {x + iy|x + y = 1} = ∅.
(A. Dorogovtsev)