Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Let {an , n ≥ 1} be any sequence of positive numbers. Denote by bn the number of terms

ak such that ak ≥ n1 . Prove that at least one of the series

∞

n=1

an and

∞

n=1

1

bn

is divergent.

(G. Shevchenko)

2. Let {Mα , α ∈ A} be a set of subsets of N such that for every α1 , α2 ∈ A we have

Mα1 ⊂ Mα2 or Mα2 ⊂ Mα1 and if α1 = α2 then Mα1 = Mα2 . Prove or disprove that A is

at most countable.

(I. Shevchuk)

3. Find all strictly increasing functions f : [0, +∞) → R such that for every x > y ≥ 0 we

(y)

have f x+y

= ff (x)+f

.

(V. Radchenko)

x−y

(x)−f (y)

4. The sequence {xn , n ≥ 1} is defined as follows: x1 = a and xn+1 = x3n − 3xn , n ≥ 1.

Determine the set of real numbers a for which the sequence is convergent. (A. Kukush)

5. Denote by dn the number of divisors of positive integer n (including 1 and n). Prove

that

∞

n=1

d(n)

n2

< 4.

(G. Shevchenko)

6. Two wolves and a hare are running at the surface of a thor {(x, y, z) | ( x2 + y 2 −

2000)2 + z 2 ≤ 2000} with speed 1. Initial distances from each wolf to the hare exceed 2000.

The wolves will catch the hare if the distance between at least one of them and the hare

became smaller then 1. The wolves and the hare see one another at any distance. Are the

wolves able to catch the hare in finite time?

(G. Shevchenko)

7. In the ring Zn of residues modulo n calculate det An , det Bn where An = (i+j)i,j=0,1,...,n−1 ,

Bn = (i · j)i,j=0,1,...,n−1 , n ≥ 2.

(V. Mazorchuk)

8. Prove that complex number z satisfies |z| − Re z ≤ 12 if and only if there exist complex

numbers u, v such that z = uv and |u − v| ≤ 1.

(V. Radchenko)

9. Two (not necessarily distinct) subsets A1 and A2 of X = {1, 2, . . . , n} are chosen in

arbitrary way. Find the probability of A1 A2 = ∅.

(M. Yadrenko)

10. There are N chairs in the first row of Room 41. If all possible ways for n persons to

chose their places are equally likely find the probability of having no persons sitting at

consequtive chairs.

(M. Yadrenko)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1

1. Compare the integrals

xx dx and

0

1 1

(xy)xy dxdy.

(V. Radchenko)

0 0

2. The sequence {xn , n ≥ 1} is defined as follows: x1 = a and xn+1 = 3xn − x3n , n ≥ 1.

Determine the set of real numbers a for which the sequence is convergent. (A. Kukush)

3. An element x of finite grope G is said to be selfdouble if there exist non necessarily

distinct elements u = e, v = e ∈ G such that x = uv = vu. Prove that if x ∈ G is not

selfdouble then x is an element of order 2 and G contains 2(2k − 1) elements for some

k ∈ N.

(V. Mazorchuk)

4. Find the number of homomorphisms of the rings Mat2×2 (C) → Mat3×3 (C) such that

the image of 2 × 2 identity matrix is 3 × 3 identity matrix.

(V. Mazorchuk)

5. Prove that the system

dx

= y 2 − xy,

dt

dy

= x4 − x3 y

dt

has no non-constant periodis solutions.

(O. Stanzhitskyy)

6. Let f be a function which is Lipshitzian in some neighborhood of zero in Rn and

→

−

→

−

f ( 0 ) = 0 . Denote by x(t, t0 , x0 ), t ≥ t0 , the solution of Cauchy problem for the system

dx

= f (x) with initial condition x(t0 ) = x0 . Prove that

dt

−

→

a) If zero solution x(t, t0 , 0 ), t ≥ t0 , is stable by Lyapunov for some t0 ∈ R then it is stable

by Lyapunov for every t0 ∈ R and uniformly over t0 .

−

→

b) If zero solution x(t, t0 , 0 ), t ≥ t0 , is asymptotically stable by Lyapunov then

lim ||x(t, t0 , x0 )|| = 0 uniformly over x0 from some neighborhood of zero in Rn .

t→+∞

∞

7. Let f : [1, +∞) → [0, +∞) be measurable function such that

is Lebesgue measure). Prove that

a) The series

∞

(O. Stanzhitskyy)

f (x)λ(dx) < ∞ (here λ

1

f (nx) converges for λ-almost all x ∈ [1, +∞).

n=1

T

1

T →+∞ T 1

b) lim

xf (x)λ(dx) = 0.

(Yu. Mishura)

8. Let ξ be nonnegative random variable. Suppose that for every x ≥ 0 the expectations

f (x) = E(ξ − x)+ ≤ ∞ are known. Find the expectation Eeξ . (y+ denotes max(y, 0).)

(A. Kukush)

9. The number of passengers at the bus stop is homogeneous Poisson process with parameter

λ which starts at time 0. The bus arrived at time t. Find the expectation of sum of waiting

times for all the passengers.

(M. Yadrenko)

10. There are N chairs in the first row of Room 41. If all possible ways for n persons to

chose their places are equally likely find the probability of having no persons sitting at

consequtive chairs.

(M. Yadrenko)

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Let {an , n ≥ 1} be any sequence of positive numbers. Denote by bn the number of terms

ak such that ak ≥ n1 . Prove that at least one of the series

∞

n=1

an and

∞

n=1

1

bn

is divergent.

(G. Shevchenko)

2. Let {Mα , α ∈ A} be a set of subsets of N such that for every α1 , α2 ∈ A we have

Mα1 ⊂ Mα2 or Mα2 ⊂ Mα1 and if α1 = α2 then Mα1 = Mα2 . Prove or disprove that A is

at most countable.

(I. Shevchuk)

3. Find all strictly increasing functions f : [0, +∞) → R such that for every x > y ≥ 0 we

(y)

have f x+y

= ff (x)+f

.

(V. Radchenko)

x−y

(x)−f (y)

4. The sequence {xn , n ≥ 1} is defined as follows: x1 = a and xn+1 = x3n − 3xn , n ≥ 1.

Determine the set of real numbers a for which the sequence is convergent. (A. Kukush)

5. Denote by dn the number of divisors of positive integer n (including 1 and n). Prove

that

∞

n=1

d(n)

n2

< 4.

(G. Shevchenko)

6. Two wolves and a hare are running at the surface of a thor {(x, y, z) | ( x2 + y 2 −

2000)2 + z 2 ≤ 2000} with speed 1. Initial distances from each wolf to the hare exceed 2000.

The wolves will catch the hare if the distance between at least one of them and the hare

became smaller then 1. The wolves and the hare see one another at any distance. Are the

wolves able to catch the hare in finite time?

(G. Shevchenko)

7. In the ring Zn of residues modulo n calculate det An , det Bn where An = (i+j)i,j=0,1,...,n−1 ,

Bn = (i · j)i,j=0,1,...,n−1 , n ≥ 2.

(V. Mazorchuk)

8. Prove that complex number z satisfies |z| − Re z ≤ 12 if and only if there exist complex

numbers u, v such that z = uv and |u − v| ≤ 1.

(V. Radchenko)

9. Two (not necessarily distinct) subsets A1 and A2 of X = {1, 2, . . . , n} are chosen in

arbitrary way. Find the probability of A1 A2 = ∅.

(M. Yadrenko)

10. There are N chairs in the first row of Room 41. If all possible ways for n persons to

chose their places are equally likely find the probability of having no persons sitting at

consequtive chairs.

(M. Yadrenko)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1

1. Compare the integrals

xx dx and

0

1 1

(xy)xy dxdy.

(V. Radchenko)

0 0

2. The sequence {xn , n ≥ 1} is defined as follows: x1 = a and xn+1 = 3xn − x3n , n ≥ 1.

Determine the set of real numbers a for which the sequence is convergent. (A. Kukush)

3. An element x of finite grope G is said to be selfdouble if there exist non necessarily

distinct elements u = e, v = e ∈ G such that x = uv = vu. Prove that if x ∈ G is not

selfdouble then x is an element of order 2 and G contains 2(2k − 1) elements for some

k ∈ N.

(V. Mazorchuk)

4. Find the number of homomorphisms of the rings Mat2×2 (C) → Mat3×3 (C) such that

the image of 2 × 2 identity matrix is 3 × 3 identity matrix.

(V. Mazorchuk)

5. Prove that the system

dx

= y 2 − xy,

dt

dy

= x4 − x3 y

dt

has no non-constant periodis solutions.

(O. Stanzhitskyy)

6. Let f be a function which is Lipshitzian in some neighborhood of zero in Rn and

→

−

→

−

f ( 0 ) = 0 . Denote by x(t, t0 , x0 ), t ≥ t0 , the solution of Cauchy problem for the system

dx

= f (x) with initial condition x(t0 ) = x0 . Prove that

dt

−

→

a) If zero solution x(t, t0 , 0 ), t ≥ t0 , is stable by Lyapunov for some t0 ∈ R then it is stable

by Lyapunov for every t0 ∈ R and uniformly over t0 .

−

→

b) If zero solution x(t, t0 , 0 ), t ≥ t0 , is asymptotically stable by Lyapunov then

lim ||x(t, t0 , x0 )|| = 0 uniformly over x0 from some neighborhood of zero in Rn .

t→+∞

∞

7. Let f : [1, +∞) → [0, +∞) be measurable function such that

is Lebesgue measure). Prove that

a) The series

∞

(O. Stanzhitskyy)

f (x)λ(dx) < ∞ (here λ

1

f (nx) converges for λ-almost all x ∈ [1, +∞).

n=1

T

1

T →+∞ T 1

b) lim

xf (x)λ(dx) = 0.

(Yu. Mishura)

8. Let ξ be nonnegative random variable. Suppose that for every x ≥ 0 the expectations

f (x) = E(ξ − x)+ ≤ ∞ are known. Find the expectation Eeξ . (y+ denotes max(y, 0).)

(A. Kukush)

9. The number of passengers at the bus stop is homogeneous Poisson process with parameter

λ which starts at time 0. The bus arrived at time t. Find the expectation of sum of waiting

times for all the passengers.

(M. Yadrenko)

10. There are N chairs in the first row of Room 41. If all possible ways for n persons to

chose their places are equally likely find the probability of having no persons sitting at

consequtive chairs.

(M. Yadrenko)