MechmatCompetition1996 97

Some problems from Competition for University Students
of Mechanics and Mathematics Faculty
of Kyiv State Taras Shevchenko University.
1996
1. Let a, b, c ∈ C. Find lim |a + b + c | .
n

n

n 1/n

n→∞

2. Let function f ∈ C([1, +∞)) be such that for every x ≥ 1 there exists a limit
Ax

lim

A→∞

ϕ(x).

f (u)du =: ϕ(x), ϕ(2) = 1 and function ϕ is continuous at point x = 1. Find
A
x

3. The function f ∈ C([0, +∞)) is such that f (x)

f 2 (u)du → 1, x → +∞. Prove that

0

1/3

1
, x → +∞.
3x
 n−1

(x
− xk ) sin(2πxk )
 k=0 k+1


 , where supremum is taken over all possible parti4. Find sup 

n−1
λ
(xk+1 − xk )2
f (x) ∼

k=0

tions of [0, 1] of the form λ = {0 = x0 < x1 < . . . < xn−1 < xn = 1}, n ≥ 1.
5. Let D be bounded connected domain with boundary ∂D and let f (z), F (z) be functions
= 0 for every z ∈ ∂D. Prove that
analytical in D. It is known that F (z) = 0 and Im Ff (z)
(z)
functions F (z) and F (z) + f (z) have equal number of zeroes in D.
6. Let A be linear operator in finite-dimensional space such that A1996 + A998 + 1996I = 0.
Prove that A has basis which consists of eigenvectors.

7. Let A1 , A2 , . . . , An+1 be n × n matrices. Prove that there exist numbers α1 , α2 , . . . , αn+1
not all of which are zeroes such that the matrix α1 A1 + . . . + αn+1 An+1 is singular.
8. Let matrix A be such that trA = 0. Prove that there exist positive integer n and matrices
A1 , . . . , An such that A = A1 + . . . + An and A2i = 0, 1 ≤ i ≤ n.
1997
1. Let 1 ≤ k < n be positive integers. Consider all possible representation of n as a sum
of two or more positive integer summands (Two representations with differ by order of
summands are assumed to be distinct). Prove that the number k appears as a summand
exactly (n − k + 3)2n−k−2 times in these representations.
2. Prove that the field Q(x) of rational functions contains two subfields F and K such that
[Q(x) : F ] < ∞. and [Q(x) : K] < ∞. but [Q(x) : (F K)] = ∞.
3. Let the matrix A ∈ Mn (C) has unique eigenvalue a. Prove that A commutes only with
polynomials of A if and only if rank(A − aI) = n − 1.
4. Let a ∈ Rm be a vector-column. Calculate (1 − aT (I + aaT )a)−1 .
+∞
5. Let f be positive non-increasing function on [1, +∞) such that 1 xf (x)dx < ∞.
+∞
f (x)
Prove that the integral
1 dx is convergent.
| sin x|1− x
1
n
j
.
6. Find lim
2 + n2
n→∞
j
j=1

7. Let I be the interval of length at least 2 and f be twice differentiable function on I such
that |f (x)| ≤ 1 and |f (x)| ≤ 1, x ∈ I. Prove that |f (x)| ≤ 2, x ∈ I.