mechmat competition2002

Open Competition for University Students
of Mechanics and Mathematics Faculty
of Kyiv State Taras Shevchenko University.
Problems for 1-2 years students.
1. Does there exist a function F : R2 → N such that the equality F (x, y) = F (y, z) holds
if and only if x = y = z?
(A. Bondarenko, A. Prymak)
sin x
cos x
2. Consider graphs of functions y = a
+a
, where a ∈ [1; 2,5]. Prove that there exists
a point M such that the distance from M to any of these graphs is less then 0,4.
(A. Kukush)
(1)
3. Consider a function f ∈ C ([−1, 1]) such that f (−1) = f (1) = 0. Prove that
∃ x ∈ [−1, 1] : f (x) = (1 + x2 )f (x).
(A. Prymak)
4. Each entry of the matrix A = (aij ) ∈ Mn (R) is equal to 0 or 1 and moreover
aii = 0, aij + aji = 1 (1 ≤ i < j ≤ n). Prove that rkA n − 1.
(A. Oliynyk)
π
2

5. Prove the inequality

(cos x)sin x
dx < 1.
(cos x)sin x + (sin x)cos x

0

(A. Kukush)
6. Find the dimension of the subspace of those linear operators ϕ on Mn (R) for which the
identity ϕ(AT ) = (ϕ(A))T holds for every matrix A ∈ Mn (R).
(A. Oliynyk)
∞
k
j
7. For every k ∈ N prove that ak =
∈
/ Q.
j!
j=1
(V. Brayman, Yu. Shelyazhenko)
8. Find all functions f ∈ C(R) such that ∀ x, y, z ∈ R holds
f (x) + f (y) + f (z) = f 37 x + 67 y − 27 z + f 67 x − 27 y + 37 z + f − 72 x + 37 y + 76 z .
(V. Brayman)
9. Construct a set A ⊂ R and a function f : A → R such that
∀ a1 , a2 ∈ A |f (a1 ) − f (a2 )| ≤ |a1 − a2 |3
and the range of f is uncountable.
(V. Brayman)
10. Prizmatoid is a convex polyhedron all the vertices of which lie in two parallel planes –
the lower and the upper bases of prizmatoid. Consider a section of a given prizmatoid by
a plane which is parallel to the bases and is at distance x from the lower base. Prove that
the area of this section is a polynomial of x of at most second degree.
(A. Kukush, R. Ushakov)


Open Competition for University Students
of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.
Problems for 3-4 years students.
1. Let ξ be a random variable with finite expectation at probability space (Ω, F, P ). Let
ω be a signed measure on F such that
∀ A ∈ F : inf ξ(x) · P (A) ≤ ω(A) ≤ sup ξ(x) · P (A).
x∈A

Prove that ∀ A ∈ F : ω(A) =

x∈A

ξ(x)dP (x).

(A. Kukush)

A

2. For every positive integer n consider function fn (x) = nsin x + ncos x , x ∈ R. Prove that
there exists a sequence {xn } such that for every n fn has a global maximum at xn and
xn → 0 as n → ∞.
(A. Kukush)
3. Let U be nonsingular real n × n matrix a ∈ Rn and let L be the subspace of Rn . Prove
that
PU T L (U −1 a) ≤ U −1 · PL a ,
where PM is a projector onto subspace M.
(A. Kukush)
4. Let f : C\{0} → (0, +∞) be continuous function such that lim f (z) = 0, lim f (z) =
z→0

|z|→∞

dz
∞. Prove that for every T > 0 there exist a solution of differential equation
= izf (z)
dt
which has period T.
(O. Stanzhitskyy)
π
2

5. Prove the inequality

(cos x)sin x
dx < 1.
(cos x)sin x + (sin x)cos x

0

(A. Kukush)
6. Find the dimension of the subspace of those linear operators ϕ on Mn (R) for which the
identity ϕ(AT ) = (ϕ(A))T holds for every matrix A ∈ Mn (R).
(A. Oliynyk)
∞
k
j
7. For every k ∈ N prove that ak =
∈
/ Q.
j!
j=1

(V. Brayman, Yu. Shelyazhenko)
8. Find all functions f ∈ C(R) such that ∀ x, y, z ∈ R holds
f (x) + f (y) + f (z) = f 37 x + 67 y − 27 z + f 67 x − 27 y + 37 z + f − 72 x + 37 y + 76 z .
(V. Brayman)
9. Construct a set A ⊂ R and a function f : A → R such that
∀ a1 , a2 ∈ A |f (a1 ) − f (a2 )| ≤ |a1 − a2 |3
and the range of f is uncountable.
(V. Brayman)
10. Prizmatoid is a convex polyhedron all the vertices of which lie in two parallel planes –
the lower and the upper bases of prizmatoid. Consider a section of a given prizmatoid by
a plane which is parallel to the bases and is at distance x from the lower base. Prove that
the area of this section is a polynomial of x of at most second degree.
(A. Kukush, R. Ushakov)