mechmat competition2007

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)