Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Prove or disprove that lim |n sin n| = +∞.

n→∞

2. Let f ∈ C 2 (R).

a) Prove that there exists θ ∈ R such that f (θ)f (θ) + 2(f (θ))2 ≥ 0.

b) Prove that there exists function G : R → R such that

(∀x ∈ R f (x)f (x) + 2(f (x))2 ≥ 0)⇐⇒ G(f (x)) is convex on R.

√ √

n

converges to some number a ∈ ( 32 4 e, 32 e).

3. Prove that the sequence an = 32 · 54 · 98 ·. . .· 2 2+1

n

4. Find all complex solutions of the system of equations

xk1 + xk2 + . . . + xkn = 0, k = 1, 2, . . . , n.

B ) then D = CA−1 B.

5. Let A be nonsingular matrix. Prove that if rankA = rank ( CA D

6. Let b(n, k) denotes the number of permutations of n elements in which just k elements

remains at their places. Calculate

n

b(n, k).

k=1

Problems for 3-4 years students.

1. Solve all complex solutions of the system of equations

xk1 + xk2 + . . . + xkn = 0, k = 1, 2, . . . , n.

2. Let A(t) be n × n matrix which is continuous on [0, +∞). Let B ⊂ Rn be the set of

initial values x(0) such that the solution x(t) of dx

= A(t)x is bounded on [0, +∞). Prove

dt

n

that B is a subspace of R and if for every f ∈ C([0, +∞), Rn ) the system

dx

= A(t)x + f (t)

(∗)

dt

has bounded on [0, +∞) solution then for every f ∈ C([0, +∞), Rn ) there exists unique

solution x(t) of (∗) which is bounded on [0, +∞) and satisfy x(0) ∈ B ⊥ . (B ⊥ denotes an

orthogonal completion of B.)

3. Let σ be arbitrary permutation of the set 1, 2, . . . , n chosen at random. (The probability

to choose each permutation is n!1 .) Find the expectation of the number of elements which

places are preserved by permutation σ.

4. Find all functions analytical in C \ {0} such that the image of any circle with center 0

belongs to some circle with center 0. (Here circle is a line.)

5. Cone in Rn is a set obtained by transition and rotation from the set {(x1 , . . . , xn ) :

x21 + . . . + x2n−1 ≤ rx2n } for some r > 0. Prove that if A is non=bounded and convex

subspace of Rn which contains no cone then there exists two-dimensional subspace B ⊂ Rn

such that projection of A to B contains no cone in R2 .

6. Let {γk , k ≥ 1} be independent standard gaussian random variables. Prove that

max1≤k≤n γk2 ln n P

:

→ 2, n → ∞.

n

2

n

k=1 γk

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Prove or disprove that lim |n sin n| = +∞.

n→∞

2. Let f ∈ C 2 (R).

a) Prove that there exists θ ∈ R such that f (θ)f (θ) + 2(f (θ))2 ≥ 0.

b) Prove that there exists function G : R → R such that

(∀x ∈ R f (x)f (x) + 2(f (x))2 ≥ 0)⇐⇒ G(f (x)) is convex on R.

√ √

n

converges to some number a ∈ ( 32 4 e, 32 e).

3. Prove that the sequence an = 32 · 54 · 98 ·. . .· 2 2+1

n

4. Find all complex solutions of the system of equations

xk1 + xk2 + . . . + xkn = 0, k = 1, 2, . . . , n.

B ) then D = CA−1 B.

5. Let A be nonsingular matrix. Prove that if rankA = rank ( CA D

6. Let b(n, k) denotes the number of permutations of n elements in which just k elements

remains at their places. Calculate

n

b(n, k).

k=1

Problems for 3-4 years students.

1. Solve all complex solutions of the system of equations

xk1 + xk2 + . . . + xkn = 0, k = 1, 2, . . . , n.

2. Let A(t) be n × n matrix which is continuous on [0, +∞). Let B ⊂ Rn be the set of

initial values x(0) such that the solution x(t) of dx

= A(t)x is bounded on [0, +∞). Prove

dt

n

that B is a subspace of R and if for every f ∈ C([0, +∞), Rn ) the system

dx

= A(t)x + f (t)

(∗)

dt

has bounded on [0, +∞) solution then for every f ∈ C([0, +∞), Rn ) there exists unique

solution x(t) of (∗) which is bounded on [0, +∞) and satisfy x(0) ∈ B ⊥ . (B ⊥ denotes an

orthogonal completion of B.)

3. Let σ be arbitrary permutation of the set 1, 2, . . . , n chosen at random. (The probability

to choose each permutation is n!1 .) Find the expectation of the number of elements which

places are preserved by permutation σ.

4. Find all functions analytical in C \ {0} such that the image of any circle with center 0

belongs to some circle with center 0. (Here circle is a line.)

5. Cone in Rn is a set obtained by transition and rotation from the set {(x1 , . . . , xn ) :

x21 + . . . + x2n−1 ≤ rx2n } for some r > 0. Prove that if A is non=bounded and convex

subspace of Rn which contains no cone then there exists two-dimensional subspace B ⊂ Rn

such that projection of A to B contains no cone in R2 .

6. Let {γk , k ≥ 1} be independent standard gaussian random variables. Prove that

max1≤k≤n γk2 ln n P

:

→ 2, n → ∞.

n

2

n

k=1 γk