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)

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)