Some problems from Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

1996

1. Let a, b, c ∈ C. Find lim |a + b + c | .

n

n

n 1/n

n→∞

2. Let function f ∈ C([1, +∞)) be such that for every x ≥ 1 there exists a limit

Ax

lim

A→∞

ϕ(x).

f (u)du =: ϕ(x), ϕ(2) = 1 and function ϕ is continuous at point x = 1. Find

A

x

3. The function f ∈ C([0, +∞)) is such that f (x)

f 2 (u)du → 1, x → +∞. Prove that

0

1/3

1

, x → +∞.

3x

n−1

(x

− xk ) sin(2πxk )

k=0 k+1

, where supremum is taken over all possible parti4. Find sup

n−1

λ

(xk+1 − xk )2

f (x) ∼

k=0

tions of [0, 1] of the form λ = {0 = x0 < x1 < . . . < xn−1 < xn = 1}, n ≥ 1.

5. Let D be bounded connected domain with boundary ∂D and let f (z), F (z) be functions

= 0 for every z ∈ ∂D. Prove that

analytical in D. It is known that F (z) = 0 and Im Ff (z)

(z)

functions F (z) and F (z) + f (z) have equal number of zeroes in D.

6. Let A be linear operator in finite-dimensional space such that A1996 + A998 + 1996I = 0.

Prove that A has basis which consists of eigenvectors.

7. Let A1 , A2 , . . . , An+1 be n × n matrices. Prove that there exist numbers α1 , α2 , . . . , αn+1

not all of which are zeroes such that the matrix α1 A1 + . . . + αn+1 An+1 is singular.

8. Let matrix A be such that trA = 0. Prove that there exist positive integer n and matrices

A1 , . . . , An such that A = A1 + . . . + An and A2i = 0, 1 ≤ i ≤ n.

1997

1. Let 1 ≤ k < n be positive integers. Consider all possible representation of n as a sum

of two or more positive integer summands (Two representations with differ by order of

summands are assumed to be distinct). Prove that the number k appears as a summand

exactly (n − k + 3)2n−k−2 times in these representations.

2. Prove that the field Q(x) of rational functions contains two subfields F and K such that

[Q(x) : F ] < ∞. and [Q(x) : K] < ∞. but [Q(x) : (F K)] = ∞.

3. Let the matrix A ∈ Mn (C) has unique eigenvalue a. Prove that A commutes only with

polynomials of A if and only if rank(A − aI) = n − 1.

4. Let a ∈ Rm be a vector-column. Calculate (1 − aT (I + aaT )a)−1 .

+∞

5. Let f be positive non-increasing function on [1, +∞) such that 1 xf (x)dx < ∞.

+∞

f (x)

Prove that the integral

1 dx is convergent.

| sin x|1− x

1

n

j

.

6. Find lim

2 + n2

n→∞

j

j=1

7. Let I be the interval of length at least 2 and f be twice differentiable function on I such

that |f (x)| ≤ 1 and |f (x)| ≤ 1, x ∈ I. Prove that |f (x)| ≤ 2, x ∈ I.

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

1996

1. Let a, b, c ∈ C. Find lim |a + b + c | .

n

n

n 1/n

n→∞

2. Let function f ∈ C([1, +∞)) be such that for every x ≥ 1 there exists a limit

Ax

lim

A→∞

ϕ(x).

f (u)du =: ϕ(x), ϕ(2) = 1 and function ϕ is continuous at point x = 1. Find

A

x

3. The function f ∈ C([0, +∞)) is such that f (x)

f 2 (u)du → 1, x → +∞. Prove that

0

1/3

1

, x → +∞.

3x

n−1

(x

− xk ) sin(2πxk )

k=0 k+1

, where supremum is taken over all possible parti4. Find sup

n−1

λ

(xk+1 − xk )2

f (x) ∼

k=0

tions of [0, 1] of the form λ = {0 = x0 < x1 < . . . < xn−1 < xn = 1}, n ≥ 1.

5. Let D be bounded connected domain with boundary ∂D and let f (z), F (z) be functions

= 0 for every z ∈ ∂D. Prove that

analytical in D. It is known that F (z) = 0 and Im Ff (z)

(z)

functions F (z) and F (z) + f (z) have equal number of zeroes in D.

6. Let A be linear operator in finite-dimensional space such that A1996 + A998 + 1996I = 0.

Prove that A has basis which consists of eigenvectors.

7. Let A1 , A2 , . . . , An+1 be n × n matrices. Prove that there exist numbers α1 , α2 , . . . , αn+1

not all of which are zeroes such that the matrix α1 A1 + . . . + αn+1 An+1 is singular.

8. Let matrix A be such that trA = 0. Prove that there exist positive integer n and matrices

A1 , . . . , An such that A = A1 + . . . + An and A2i = 0, 1 ≤ i ≤ n.

1997

1. Let 1 ≤ k < n be positive integers. Consider all possible representation of n as a sum

of two or more positive integer summands (Two representations with differ by order of

summands are assumed to be distinct). Prove that the number k appears as a summand

exactly (n − k + 3)2n−k−2 times in these representations.

2. Prove that the field Q(x) of rational functions contains two subfields F and K such that

[Q(x) : F ] < ∞. and [Q(x) : K] < ∞. but [Q(x) : (F K)] = ∞.

3. Let the matrix A ∈ Mn (C) has unique eigenvalue a. Prove that A commutes only with

polynomials of A if and only if rank(A − aI) = n − 1.

4. Let a ∈ Rm be a vector-column. Calculate (1 − aT (I + aaT )a)−1 .

+∞

5. Let f be positive non-increasing function on [1, +∞) such that 1 xf (x)dx < ∞.

+∞

f (x)

Prove that the integral

1 dx is convergent.

| sin x|1− x

1

n

j

.

6. Find lim

2 + n2

n→∞

j

j=1

7. Let I be the interval of length at least 2 and f be twice differentiable function on I such

that |f (x)| ≤ 1 and |f (x)| ≤ 1, x ∈ I. Prove that |f (x)| ≤ 2, x ∈ I.

## Kiểm tra văn 6 tiết 97

## de thi tuyen vao 10 TP ha nam 96-97

## Tiet 97; Phan so va phep chia tu nhien

## Tiết 97, 98: Người trong bao

## Unit 9: Lesson 1: A1-2/ p96-97

## Unit 9: Lesson 2: A3- 5/p.97-98

## Tiết 97: Người trong bao

## Tiết 97: Người trong bao

## Tiết 97: Nước Đại việt ta

## 97 câu trắc nghiệm về ý nghĩa của đạo hàm

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