Open Competition for University Students

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 1-2 years students

n

(k + p)(k + q)

.

n→∞

(k

+

r)(k

+

s)

k=1

1. Let p, q, r, s be positive integers. Find the limit lim

n

2. Is it true that for every n

k(2n

k ) is divisible by 8?

2 the number

k=1

3. Two players in turn replace asterisks in the matrix

∗

∗

...

∗

∗

∗

...

∗

...

...

...

...

∗

∗

...

∗

(R. Ushakov)

(A. Kukush)

of size 10 × 10 by positive integers

1, . . . , 100 (at each turn one may take any number which has not been used earlier and replace by

it any asterisk). If they form a non-singular matrix then the first player wins, else the second player

wins. Has anybody of players a winning strategy? If somebody has, then who?

(V. Brayman)

1

4. Prove that a function f ∈ C (0, +∞) which satisfy

1

f (x) =

, x > 0,

4

1 + x + cos f (x)

is bounded at (0, +∞).

(O. Nesterenko)

5. Does there exist a polynomial, which takes value k exactly at k distinct real points for every

1 k 2007?

(V. Brayman)

6. The clock-face is a dice of radius 1. The hour-hand is a dice of radius 1/2 touching the circle of

the clock-face in inner way, and the minute-hand is a segment of length 1. Find the area of the figure

formed by all intersections of hands in 12 hours (i.e. in one full turn of the hour-hand).

(G. Shevchenko)

7. Find the maximum of x31 + . . . + x310 for x1 , . . . , x10 ∈ [−1, 2] such that x1 + . . . + x10 = 10.

(D. Mitin)

8. Let a0 = 1, a1 = 1 and an = an−1 + (n − 1)an−2 , n 2. Prove that for every odd number p the

number ap − 1 is divisible by p.

(O. Rybak)

9. Find all positive integers n for which there exist infinitely many matrices A of size n × n with

integer entries such that An = I (here I is the identity matrix).

(A. Bondarenko, M. Vyazovska)

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. Does the Riemann integral

0

sin x dx

converge?

x + ln x

(A. Kukush)

2. The clock-face is a dice of radius 1. The hour-hand is a dice of radius 1/2 touching the circle of

the clock-face in inner way, and the minute-hand is a segment of length 1. Find the area of the figure

formed by all intersections of hands in 12 hours (i.e. in one full turn of the hour-hand).

(G. Shevchenko)

1

3. Prove that a function f ∈ C (0, +∞) which satisfy

1

, x > 0,

f (x) =

4

1 + x + cos f (x)

is bounded at (0, +∞).

(O. Nesterenko)

4. Does there exist a polynomial, which takes value k exactly at k distinct real points for every

1 k 2007?

(V. Brayman)

5. Let f : R → [0, +∞) be measurable function such that A f dλ < +∞ for every set A of finite

Lebesgue measure (i.e. λ(A) < +∞). Prove that there exist a constant M and Lebesgue integrable

function g : R → [0, +∞) such that f (x) g(x) + M, x ∈ R.

(V. Radchenko)

1

6. Investigate the character of monotonicity of a function f (σ) = E

, σ > 0, where ξ is a

1 + eξ

gaussian random variable with mean m and covariance σ 2 (m is a real parameter).

(A. Kukush)

3

3

7. Find the maximum of x1 + . . . + x10 for x1 , . . . , x10 ∈ [−1, 2] such that x1 + . . . + x10 = 10.

(D. Mitin)

8. Let A, B be symmetric real positively defined matrices and the matrix A + B − E is positively

defined as well. Is it possible that the matrix A−1 +B −1 − 21 (A−1 B −1 +B −1 A−1 ) is negatively defined?

(A. Kukush)

9. Let P (z) be polynomial with leading coefficient 1. Prose that there exists a point z0 at the unit

circle {z ∈ C : |z| = 1} such that |P (z0 )| 1.

(O. Rybak)

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 1-2 years students

n

(k + p)(k + q)

.

n→∞

(k

+

r)(k

+

s)

k=1

1. Let p, q, r, s be positive integers. Find the limit lim

n

2. Is it true that for every n

k(2n

k ) is divisible by 8?

2 the number

k=1

3. Two players in turn replace asterisks in the matrix

∗

∗

...

∗

∗

∗

...

∗

...

...

...

...

∗

∗

...

∗

(R. Ushakov)

(A. Kukush)

of size 10 × 10 by positive integers

1, . . . , 100 (at each turn one may take any number which has not been used earlier and replace by

it any asterisk). If they form a non-singular matrix then the first player wins, else the second player

wins. Has anybody of players a winning strategy? If somebody has, then who?

(V. Brayman)

1

4. Prove that a function f ∈ C (0, +∞) which satisfy

1

f (x) =

, x > 0,

4

1 + x + cos f (x)

is bounded at (0, +∞).

(O. Nesterenko)

5. Does there exist a polynomial, which takes value k exactly at k distinct real points for every

1 k 2007?

(V. Brayman)

6. The clock-face is a dice of radius 1. The hour-hand is a dice of radius 1/2 touching the circle of

the clock-face in inner way, and the minute-hand is a segment of length 1. Find the area of the figure

formed by all intersections of hands in 12 hours (i.e. in one full turn of the hour-hand).

(G. Shevchenko)

7. Find the maximum of x31 + . . . + x310 for x1 , . . . , x10 ∈ [−1, 2] such that x1 + . . . + x10 = 10.

(D. Mitin)

8. Let a0 = 1, a1 = 1 and an = an−1 + (n − 1)an−2 , n 2. Prove that for every odd number p the

number ap − 1 is divisible by p.

(O. Rybak)

9. Find all positive integers n for which there exist infinitely many matrices A of size n × n with

integer entries such that An = I (here I is the identity matrix).

(A. Bondarenko, M. Vyazovska)

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. Does the Riemann integral

0

sin x dx

converge?

x + ln x

(A. Kukush)

2. The clock-face is a dice of radius 1. The hour-hand is a dice of radius 1/2 touching the circle of

the clock-face in inner way, and the minute-hand is a segment of length 1. Find the area of the figure

formed by all intersections of hands in 12 hours (i.e. in one full turn of the hour-hand).

(G. Shevchenko)

1

3. Prove that a function f ∈ C (0, +∞) which satisfy

1

, x > 0,

f (x) =

4

1 + x + cos f (x)

is bounded at (0, +∞).

(O. Nesterenko)

4. Does there exist a polynomial, which takes value k exactly at k distinct real points for every

1 k 2007?

(V. Brayman)

5. Let f : R → [0, +∞) be measurable function such that A f dλ < +∞ for every set A of finite

Lebesgue measure (i.e. λ(A) < +∞). Prove that there exist a constant M and Lebesgue integrable

function g : R → [0, +∞) such that f (x) g(x) + M, x ∈ R.

(V. Radchenko)

1

6. Investigate the character of monotonicity of a function f (σ) = E

, σ > 0, where ξ is a

1 + eξ

gaussian random variable with mean m and covariance σ 2 (m is a real parameter).

(A. Kukush)

3

3

7. Find the maximum of x1 + . . . + x10 for x1 , . . . , x10 ∈ [−1, 2] such that x1 + . . . + x10 = 10.

(D. Mitin)

8. Let A, B be symmetric real positively defined matrices and the matrix A + B − E is positively

defined as well. Is it possible that the matrix A−1 +B −1 − 21 (A−1 B −1 +B −1 A−1 ) is negatively defined?

(A. Kukush)

9. Let P (z) be polynomial with leading coefficient 1. Prose that there exists a point z0 at the unit

circle {z ∈ C : |z| = 1} such that |P (z0 )| 1.

(O. Rybak)