mechmat competition2006

Open Competition for University Students
of The Faculty of Mechanics and Mathematics
of Kyiv National Taras Shevchenko University
Problems for 1-2 years students
1. Find all positive integers n such that the polinomial (x4 − 1)n + (x2 − x)n is divisible by x5 − 1.
(A. Kukush)
2. Let z ∈ C be such that points z 3 , 2z 3 + z 2 , 3z 3 + 3z 2 + z and 4z 3 + 6z 2 + 4z + 1 are the vertices
of an inscribed quadrangle at complex plane. Find Re z.
(V. Brayman)
n
3. Find the minimum over all unit vectors x1 , . . . , xn+1 ∈ R of max (xi , xj ).
1 i
m×n

(A. Bondarenko, M. Vyazovska)
and let B be symmetric n × n matrix such that

4. Let Em be m × m identity matrix, A ∈ R
Em A
is positively defined. Prove that “matrix determinant” B − AT A is also
block matrix
AT B
positively defined. (Symmetric matrix M is said to be positively defined if for every vector-column
x = 0 the inequality xT M x > 0 holds.)
(A. Kukush)
5. Does there exist an infinite set of square symmetric matrices M such that for any distinct matrices
A, B ∈ M AB 2 = B 2 A holds but AB = BA?
(V. Brayman)
f (x)
6. Let f : (0, +∞) → R be continuous concave function, lim f (x) = +∞, lim

= 0. Prove
x→+∞
x→+∞ x
that sup {f (n)} = 1, where {a} = a − [a] is fractional part of a.
n∈N

(O. Nesterenko)

7. Does there exist a continuous function f : R → (0, 1) such that the sequence
n

n

f (x) dx, n

an =
−n

1, converges and the sequence bn =

f (x) ln f (x) dx, n

1, is divergent?

−n

(A. Kukush)
8. Let P (x) be a polinomial such that there exist infinitely many pairs of integers (a, b) such that
P (a + 3b) + P (5a + 7b) = 0. Prove that the polinomial P (x) has an integer root.
(V. Brayman)

n
ai aj
9. For every real numbers a1 , a2 , . . . , an ∈ R \ {0} prove the inequality
0.
2
a + a2j
i,j=1 i
(S. Novak, Great Britain)


Open Competition for University Students
of The Faculty of Mechanics and Mathematics
of Kyiv National Taras Shevchenko University
Problems for 3-4 years students
1. Let z ∈ C be such that points z 3 , 2z 3 + z 2 , 3z 3 + 3z 2 + z and 4z 3 + 6z 2 + 4z + 1 are the vertices
of an inscribed quadrangle at complex plane. Find Re z.
(V. Brayman)
∞

2. Does there exist a continuous function f : R → (0, 1) such that
∞

while

f (x) dx < ∞
−∞

f (x) ln f (x) dx is divergent?
−∞
m×n

(A. Kukush)
and let B be symmetric n × n matrix such that

3. Let Em be m × m identity matrix, A ∈ R
Em A
block matrix
is positively defined. Prove that “matrix determinant” B − AT A is also
AT B
positively defined.
(A. Kukush)
4. Does there exist an infinite set of square symmetric matrices M such that for any distinct matrices
A, B ∈ M holds AB 2 = B 2 A but AB = BA?
(V. Brayman)
5. a) Let ξ and η be random variables (not necessarily independent) which have continuous distribution functions. Prove that min(ξ, η) also has continuous distribution function.
b) Let ξ and η be random variables which have densities. Is it true that min(ξ, η) also has a density?
(A. Kukush, G. Shevchenko)
6. Is it possible to choose uncountable set A ⊂ l2 of elements with unit norm such that for any
∞

distinct x = (x1 , . . . , xn , . . .), y = (y1 , . . . , yn , . . .) from A the series

|xn − yn | is divergent?
n=1

(A. Bondarenko)
7. Let ξ, η be independent identically distributed random variables such that
ξη
0.
P (ξ = 0) = 1. Prove the inequality E 2

ξ + η2
(S. Novak, Great Britain)
8. For every n ∈ N find the minimal λ > 0 such that for every convex compact set K ⊂ Rn there
exist a point x ∈ K such that the set which is homothetic to K with centre x and coefficient (−λ)
contains K.
(O. Lytvak, Canada)
9. Let X = L1 [0, 1] and let Tn : X → X be the sequence of nonnegative (i.e. f
0 =⇒ Tn f
0)
linear operators such that Tn
1 and lim f − Tn f X = 0 for f (x) ≡ x and for f (x) ≡ 1. Prove
that lim f − Tn f
n→∞

n→∞

X

= 0 for every f ∈ X.

(A. Prymak)