Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems 1-8 are for 1-2 years students, problems 5-12 are for 3-4 years students.

∞

9n + 4

.

n(3n

+

1)(3n

+

2)

n=1

√

√

2. Evaluate the limit lim N 1 − max { n} ,

(A. Kukush)

1. Evaluate

N →∞

1≤n≤N

where {x} denotes fractional part of x.

(D. Mitin)

3. For every n ∈ N find the minimum of k ∈ N for which there exist x1 , . . . , xk ∈ Rn such

k

n

that ∀ x ∈ R ∃ a1 , . . . , ak > 0 : x =

ai xi .

(A. Bondarenko)

i=1

4. Find all n ∈ N for which there exist square n × n matrices A and B such that rankA +

rankB ≤ n and every square real matrix X which commutes with A and B is of the form

X = λI, λ ∈ R.

(A. Bondarenko)

√

(A. Kukush)

5. Prove the inequality 2 3 3 4 4 . . . n n < 2, n 2.

∞

n

n

2

x3 + x3

6. For every real x = 1 find the sum of the series

.

3n+1

1

−

x

n=0

7. For every positive integers m ≤ n prove the inequality

m

n

k

m!

k

(−1)m+k Cm

≤ Cnm m .

m

m

k=0

(A. Kukush)

(D. Mitin)

8. A parabola with focus F and a triangle T are given at the plane. Construct with the

compass and the ruler a triangle similar to T such that one of its vertices is F and two

other vertices lie on parabola.

(G. Shevchenko)

2

9. Do there exist a set A ⊂ R , measurable by Lebesgue such that for every set E with

zero Lebesgue measure the set A\E is not Borelian?

(A. Bondarenko)

10. Given is a real symmetric matrix A = (aij )ni,j=1 with eigenvectors ek , k = 1, n and

eigenvalues λk , k = 1, n respectively. Construct a real symmetric nonnegatively definite

matrix X = (xij )ni,j=1 which minimizes the distance d(X, A) =

n

(xij − aij )2 .

i,j=1

(A. Kukush)

11. Let ϕ be a conform mapping from Ω = {Imz > 0}\T onto {Imz > 0}, where T is a

triangle with vertices {1, −1, i}. Prove that if z0 ∈ Ω and ϕ(z0 ) = z0 then |ϕ (z0 )| 1.

(T. Androshchuk)

12. The vertices of a triangle are independent uniformly distributed at unit circle random

points. Find the expectation of the area of this triangle.

(A. Kukush)

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems 1-8 are for 1-2 years students, problems 5-12 are for 3-4 years students.

∞

9n + 4

.

n(3n

+

1)(3n

+

2)

n=1

√

√

2. Evaluate the limit lim N 1 − max { n} ,

(A. Kukush)

1. Evaluate

N →∞

1≤n≤N

where {x} denotes fractional part of x.

(D. Mitin)

3. For every n ∈ N find the minimum of k ∈ N for which there exist x1 , . . . , xk ∈ Rn such

k

n

that ∀ x ∈ R ∃ a1 , . . . , ak > 0 : x =

ai xi .

(A. Bondarenko)

i=1

4. Find all n ∈ N for which there exist square n × n matrices A and B such that rankA +

rankB ≤ n and every square real matrix X which commutes with A and B is of the form

X = λI, λ ∈ R.

(A. Bondarenko)

√

(A. Kukush)

5. Prove the inequality 2 3 3 4 4 . . . n n < 2, n 2.

∞

n

n

2

x3 + x3

6. For every real x = 1 find the sum of the series

.

3n+1

1

−

x

n=0

7. For every positive integers m ≤ n prove the inequality

m

n

k

m!

k

(−1)m+k Cm

≤ Cnm m .

m

m

k=0

(A. Kukush)

(D. Mitin)

8. A parabola with focus F and a triangle T are given at the plane. Construct with the

compass and the ruler a triangle similar to T such that one of its vertices is F and two

other vertices lie on parabola.

(G. Shevchenko)

2

9. Do there exist a set A ⊂ R , measurable by Lebesgue such that for every set E with

zero Lebesgue measure the set A\E is not Borelian?

(A. Bondarenko)

10. Given is a real symmetric matrix A = (aij )ni,j=1 with eigenvectors ek , k = 1, n and

eigenvalues λk , k = 1, n respectively. Construct a real symmetric nonnegatively definite

matrix X = (xij )ni,j=1 which minimizes the distance d(X, A) =

n

(xij − aij )2 .

i,j=1

(A. Kukush)

11. Let ϕ be a conform mapping from Ω = {Imz > 0}\T onto {Imz > 0}, where T is a

triangle with vertices {1, −1, i}. Prove that if z0 ∈ Ω and ϕ(z0 ) = z0 then |ϕ (z0 )| 1.

(T. Androshchuk)

12. The vertices of a triangle are independent uniformly distributed at unit circle random

points. Find the expectation of the area of this triangle.

(A. Kukush)