Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Prove that for every positive integer n an inequality

1

3

2n − 1

1

+ + ... +

<

3! 4!

(n + 2)!

2

holds.

(R. Ushakov)

2. One cell is erased from the 2 × n table in arbitrary way. Find the probability of the

following event: It is possible to cover without overlappings the rest of the table with figures

(with any orientation).

3. For any continuous convex at [0, 1] function f prove the inequality

4/5

2 1

2 3/5

f (x) dx +

f (x) dx

f (x) dx.

5 0

3 0

0

(A. Kukush)

(A. Prymak)

4. Find all odd continuous functions f : R → R such that the identity f (f (x)) = x holds

for every real x.

(A. Kukush)

5. Construct with the compass and the ruler the circle of maximal radius which lies inside

the given parabola and touches to it in its vertice.

(Zh. Chernousova)

6. Let A, B, C, D be (not necessarily square) real matrices such that

AT = BCD, B T = CDA, C T = DAB, DT = ABC.

For S = ABCD prove that S 3 = S.

(V. Brayman)

7. Denote by An the maximum of the determinant of an n × n matrix with entries ±1.

Does there exist a finite limit lim n An ?

(A. Bondarenko)

n→∞

8. Let {xn , n 1} be a sequence of positive numbers which contains at least two distinct

√

elements. Prove or disprove that lim (x1 + · · · + xn − n n x1 . . . xn ) > 0.

(D. Mitin)

n→∞

9. Some permutation of entries of a matrix maps any non-singular n × n matrix into nonsingular one and maps identity matrix into itself. Prove that this permutation preserves

the determinant of the matrix.

(V. Brayman)

10. A rectangle with side lengths a0 , b0 is dissected into smaller rectangulars with side

lengths ak , bk , 1 ≤ k ≤ n. The sides of smaller rectangulars are parallel to the corresponding

sides of the big rectangular. Prove that

| sin a0 sin b0 | ≤ nk=1 | sin ak sin bk |.

(Zsolt P´ales, Hungary)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1. Let random variable ξ be distributed as |γ|α , α ∈ R, where γ is standard normal variable.

For which α there exists an expectation M ξ?

(U. Mishura)

2. One cell is erased from the 2 × n table in arbitrary way. Find the probability of the

following event: It is possible to cover without overlappings the rest of the table with figures

(with any orientation).

(A. Kukush)

3. Normed space Y is said to be strongly normed if for every y1 , y2 ∈ Y the equality

y1 + y2

y1 = y2 =

implies y1 = y2 . Let X be normed space, let G be subspace of X

2

and let the adjoint space X ∗ be strongly normed. Prove that for every functional from G∗

there exists the unique extension X ∗ which preserves the norm.

(O. Nesterenko)

z2

4. Let R(z) =

− z + ln(1 + z), z ∈ C, z = −1. Prove that for every real x the following

2

|x|3

.

inequality holds: |R(ix)| ≤

3

(“ln” means the value of logarithm from the branch with ln 1 = 0.)

(D. Mitin)

5. Let A, B, C, D be (not necessarily square) real matrices such that

AT = BCD, B T = CDA, C T = DAB, DT = ABC.

For S = ABCD prove that S 2 = S.

Remark: for 1-2 years students it is proposed to prove that S 3 = S.

(V. Brayman)

6. Let e be nonzero vector in R2 . Construct a nonsingular matrix A ∈ R2×2 such that for

fd (x) := A(x + d) 2 , x, d ∈ R2 the set of pairs M := {(x, y) : fe (x) = 1, f−e (y) = 1 and

there exist real numbers λ, µ such that (x, y) is a stationary point of Lagrange function

F (x, y) := x − y 2 + λfe (x) + µf−e (y)} contains at least 8 pairs of points. (A. Kukush)

7. Some permutation of entries of a matrix maps any non-singular n × n matrix into nonsingular one and maps identity matrix into itself. Prove that this permutation preserves

the determinant of the matrix.

(V. Brayman)

8. The croupier and two players are playing the following game. The croupier chooses an

integer in the interval [1, 2004] with uniform probability. The players guess the integer in

turn. After each guess the croupier informs them whether the chosen integer is bigger or

smaller or has just been guessed. The player who guesses the integer first wins. Prove that

both players have a strategy such that their chances of winning are at least 21 .

(S. Shklyar)

9. A rectangle with side lengths a0 , b0 is dissected into smaller rectangulars with side lengths

ak , bk , 1 ≤ k ≤ n. The sides of smaller rectangulars are parallel to the corresponding sides

of the big rectangular. Prove that

| sin a0 sin b0 | ≤ nk=1 | sin ak sin bk |.

(Zsolt P´ales, Hungary)

10. Does there exist a sequence {xn , n 1} of vectors from l2 with unit norm which satisfy

1

for n = m, n, m ∈ N?

(A. Bondarenko)

(xn , xm ) < −

2004

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Prove that for every positive integer n an inequality

1

3

2n − 1

1

+ + ... +

<

3! 4!

(n + 2)!

2

holds.

(R. Ushakov)

2. One cell is erased from the 2 × n table in arbitrary way. Find the probability of the

following event: It is possible to cover without overlappings the rest of the table with figures

(with any orientation).

3. For any continuous convex at [0, 1] function f prove the inequality

4/5

2 1

2 3/5

f (x) dx +

f (x) dx

f (x) dx.

5 0

3 0

0

(A. Kukush)

(A. Prymak)

4. Find all odd continuous functions f : R → R such that the identity f (f (x)) = x holds

for every real x.

(A. Kukush)

5. Construct with the compass and the ruler the circle of maximal radius which lies inside

the given parabola and touches to it in its vertice.

(Zh. Chernousova)

6. Let A, B, C, D be (not necessarily square) real matrices such that

AT = BCD, B T = CDA, C T = DAB, DT = ABC.

For S = ABCD prove that S 3 = S.

(V. Brayman)

7. Denote by An the maximum of the determinant of an n × n matrix with entries ±1.

Does there exist a finite limit lim n An ?

(A. Bondarenko)

n→∞

8. Let {xn , n 1} be a sequence of positive numbers which contains at least two distinct

√

elements. Prove or disprove that lim (x1 + · · · + xn − n n x1 . . . xn ) > 0.

(D. Mitin)

n→∞

9. Some permutation of entries of a matrix maps any non-singular n × n matrix into nonsingular one and maps identity matrix into itself. Prove that this permutation preserves

the determinant of the matrix.

(V. Brayman)

10. A rectangle with side lengths a0 , b0 is dissected into smaller rectangulars with side

lengths ak , bk , 1 ≤ k ≤ n. The sides of smaller rectangulars are parallel to the corresponding

sides of the big rectangular. Prove that

| sin a0 sin b0 | ≤ nk=1 | sin ak sin bk |.

(Zsolt P´ales, Hungary)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1. Let random variable ξ be distributed as |γ|α , α ∈ R, where γ is standard normal variable.

For which α there exists an expectation M ξ?

(U. Mishura)

2. One cell is erased from the 2 × n table in arbitrary way. Find the probability of the

following event: It is possible to cover without overlappings the rest of the table with figures

(with any orientation).

(A. Kukush)

3. Normed space Y is said to be strongly normed if for every y1 , y2 ∈ Y the equality

y1 + y2

y1 = y2 =

implies y1 = y2 . Let X be normed space, let G be subspace of X

2

and let the adjoint space X ∗ be strongly normed. Prove that for every functional from G∗

there exists the unique extension X ∗ which preserves the norm.

(O. Nesterenko)

z2

4. Let R(z) =

− z + ln(1 + z), z ∈ C, z = −1. Prove that for every real x the following

2

|x|3

.

inequality holds: |R(ix)| ≤

3

(“ln” means the value of logarithm from the branch with ln 1 = 0.)

(D. Mitin)

5. Let A, B, C, D be (not necessarily square) real matrices such that

AT = BCD, B T = CDA, C T = DAB, DT = ABC.

For S = ABCD prove that S 2 = S.

Remark: for 1-2 years students it is proposed to prove that S 3 = S.

(V. Brayman)

6. Let e be nonzero vector in R2 . Construct a nonsingular matrix A ∈ R2×2 such that for

fd (x) := A(x + d) 2 , x, d ∈ R2 the set of pairs M := {(x, y) : fe (x) = 1, f−e (y) = 1 and

there exist real numbers λ, µ such that (x, y) is a stationary point of Lagrange function

F (x, y) := x − y 2 + λfe (x) + µf−e (y)} contains at least 8 pairs of points. (A. Kukush)

7. Some permutation of entries of a matrix maps any non-singular n × n matrix into nonsingular one and maps identity matrix into itself. Prove that this permutation preserves

the determinant of the matrix.

(V. Brayman)

8. The croupier and two players are playing the following game. The croupier chooses an

integer in the interval [1, 2004] with uniform probability. The players guess the integer in

turn. After each guess the croupier informs them whether the chosen integer is bigger or

smaller or has just been guessed. The player who guesses the integer first wins. Prove that

both players have a strategy such that their chances of winning are at least 21 .

(S. Shklyar)

9. A rectangle with side lengths a0 , b0 is dissected into smaller rectangulars with side lengths

ak , bk , 1 ≤ k ≤ n. The sides of smaller rectangulars are parallel to the corresponding sides

of the big rectangular. Prove that

| sin a0 sin b0 | ≤ nk=1 | sin ak sin bk |.

(Zsolt P´ales, Hungary)

10. Does there exist a sequence {xn , n 1} of vectors from l2 with unit norm which satisfy

1

for n = m, n, m ∈ N?

(A. Bondarenko)

(xn , xm ) < −

2004