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)

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)