mechmat competition2003

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)