Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

The numbers in braces denotes for which years students the problems are proposed.

1.(1) Prove that for every n ∈ N there exists unique t(n) > 0 such that (t(n)−1) ln t(n) = n.

Calculate lim t(n) lnnn .

n→∞

2.(1) Let {an , n ≥ 1} ⊂ R be bounded sequence. Define An = 1] n(a1 + . . . + an ) n ≥ 1.

Assume that the set A of partial limits of {an , n ≥ 1} coincides with the set of partial

limits of {An , n ≥ 1}. Prove that A is a segment or a single point. Prove or disprove

that if A is a segment or a single point then the sets of partial limits of {an , n ≥ 1} and

{An , n ≥ 1} coincide.

3.(1) Let f : R → R has primitive F on R and satisfy 2xF (x) = f (x), x ∈ R. Find F.

c

f (x)dx = (1 − c)f (c).

4.(1) Let f ∈ C([0, 1]). Prove that there exists c ∈ (0, 1) such that

0

5.(1-2) The sequence of m × m real matrices {An , n ≥ 0} is defined as follows: A0 = A,

An+1 = A2n − An + 43 I, n ≥ 0, where A is positively defined matrix such that tr(A) < 1.

Find lim An .

n→∞

6.(1-2) Let {xn , n ≥ 1} ⊂ R be bounded sequence and a be real number such that

n

1

xj

n→∞ n k=1 k

lim

n

1

sin xk

n→∞ n k=1

= aj , j = 1, 2. Prove that lim

= sin a.

7.(1-4) Let F be any quadrangle with area 1 and G be a disk with radius π1 . For every

n ≥ 1 let a(n) be the maximum number of figures of area n1 similar to F without common

interior points which is possible to pack into G. Similarly define b(n) as the maximum

number of disks of area n1 without common interior points which is possible to pack into

b(n)

a(n)

F. Prove that lim

< lim

= 1.

n→∞ n

n→∞ n

8.(1-4) Find the maximum of perimeter of convex piecewise-smooth closed curve with

diameter d.

9.(2) Prove that the equation y (x)−(2+cos x)y(x) = arctan x, x ∈ R, has unique bounded

on R solution in C 1 (R).

x

y (x) = 0 sin(y(x))du + cos x, x ≥ 0,

10.(2) Solve

y(0) = 0.

11.(3) The series f (z) =

∞

cn z n has radius of convergence 1. It is known that cn = 0 for

n=0

n = km + l, m ∈ N, k ≥ 2. Prove that f has at least two singular points at unit circle.

12.(3-4) Let K = {z ∈ C | 1 ≤ |z| ≤ 2}. Consider the set W of functions u which are

∂u

ds = 2π, Sj = {z ∈ C | |z| = j}, j = 1, 2, n is normal to Sj

harmonic in K and satisfy ∂n

Sj

∗

inside the K. Let u ∈ W be such that D(u∗ ) = min D(u), where D(u) =

u∈W

K

(u 2x +u 2y )dxdy.

Prove that u∗ is constant at S1 and at S2 .

13.(3-4) Each positive integer is a trap with probability 0.4 independently from other

integers. A hare is jumping over positive integers. It starts from 1 and jumps each time at

distance 0,1 or 2 to the right with probability 31 and independently from trap’s positions

and from previous jumps. Prove that the hare eventually will get into trap with probability

1.

14.(4) Let H be Hilbert space and An , n ≥ 1 be linear operators such that for every x ∈ H

we have An x → ∞, n → ∞. Prove that we have An K → ∞, n → ∞ holds for every

compact operator K.

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

The numbers in braces denotes for which years students the problems are proposed.

1.(1) Prove that for every n ∈ N there exists unique t(n) > 0 such that (t(n)−1) ln t(n) = n.

Calculate lim t(n) lnnn .

n→∞

2.(1) Let {an , n ≥ 1} ⊂ R be bounded sequence. Define An = 1] n(a1 + . . . + an ) n ≥ 1.

Assume that the set A of partial limits of {an , n ≥ 1} coincides with the set of partial

limits of {An , n ≥ 1}. Prove that A is a segment or a single point. Prove or disprove

that if A is a segment or a single point then the sets of partial limits of {an , n ≥ 1} and

{An , n ≥ 1} coincide.

3.(1) Let f : R → R has primitive F on R and satisfy 2xF (x) = f (x), x ∈ R. Find F.

c

f (x)dx = (1 − c)f (c).

4.(1) Let f ∈ C([0, 1]). Prove that there exists c ∈ (0, 1) such that

0

5.(1-2) The sequence of m × m real matrices {An , n ≥ 0} is defined as follows: A0 = A,

An+1 = A2n − An + 43 I, n ≥ 0, where A is positively defined matrix such that tr(A) < 1.

Find lim An .

n→∞

6.(1-2) Let {xn , n ≥ 1} ⊂ R be bounded sequence and a be real number such that

n

1

xj

n→∞ n k=1 k

lim

n

1

sin xk

n→∞ n k=1

= aj , j = 1, 2. Prove that lim

= sin a.

7.(1-4) Let F be any quadrangle with area 1 and G be a disk with radius π1 . For every

n ≥ 1 let a(n) be the maximum number of figures of area n1 similar to F without common

interior points which is possible to pack into G. Similarly define b(n) as the maximum

number of disks of area n1 without common interior points which is possible to pack into

b(n)

a(n)

F. Prove that lim

< lim

= 1.

n→∞ n

n→∞ n

8.(1-4) Find the maximum of perimeter of convex piecewise-smooth closed curve with

diameter d.

9.(2) Prove that the equation y (x)−(2+cos x)y(x) = arctan x, x ∈ R, has unique bounded

on R solution in C 1 (R).

x

y (x) = 0 sin(y(x))du + cos x, x ≥ 0,

10.(2) Solve

y(0) = 0.

11.(3) The series f (z) =

∞

cn z n has radius of convergence 1. It is known that cn = 0 for

n=0

n = km + l, m ∈ N, k ≥ 2. Prove that f has at least two singular points at unit circle.

12.(3-4) Let K = {z ∈ C | 1 ≤ |z| ≤ 2}. Consider the set W of functions u which are

∂u

ds = 2π, Sj = {z ∈ C | |z| = j}, j = 1, 2, n is normal to Sj

harmonic in K and satisfy ∂n

Sj

∗

inside the K. Let u ∈ W be such that D(u∗ ) = min D(u), where D(u) =

u∈W

K

(u 2x +u 2y )dxdy.

Prove that u∗ is constant at S1 and at S2 .

13.(3-4) Each positive integer is a trap with probability 0.4 independently from other

integers. A hare is jumping over positive integers. It starts from 1 and jumps each time at

distance 0,1 or 2 to the right with probability 31 and independently from trap’s positions

and from previous jumps. Prove that the hare eventually will get into trap with probability

1.

14.(4) Let H be Hilbert space and An , n ≥ 1 be linear operators such that for every x ∈ H

we have An x → ∞, n → ∞. Prove that we have An K → ∞, n → ∞ holds for every

compact operator K.

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