Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Is it true that the sequence {xn , n ≥ 1} of real numbers is convergent if and only if

lim lim |xn − xm | = 0?

(O. Nesterenko)

n→∞ m→∞

2. Let A, B, C be real matrices. Prove the inequality

tr(A(AT − B T ) + B(B T − C T ) + C(C T − AT )) ≥ 0.

(M. Vyazovska)

3. Billiard table is obtained by cutting some squares of a chessboard. Billiard ball is

shot from one of the table corners such that its trajectory forms angle α with the side of

the billiard table, tg α ∈ Q. When the ball hit the border of the billiard table it reflects

according to the rule: the angle of incidence equals angle of reflection. If the ball landing

in any corner it falls into a hole. Prove that the ball will necessarily fall into a hole.

(G. Kryukova)

4. Solve an equation lim

n→∞

1+

x+

x2 + · · · +

√

xn = 2.

(A. Kukush)

5. Do there exist matrices A, B, C which have no common eigenvectors and satisfy the

condition AB = BC = CA?

(V. Brayman)

π

6. Prove that

7. Let f ∈ C

cos 2x cos 3x cos 4x . . . cos 2005x dx > 0.

−π

(1)

(M. Pupashenko)

(R) and a1 < a2 < a3 < b1 < b2 < b3 . Prove or disprove that there exist

f (bi ) − f (ai )

real numbers c1 ≤ c2 ≤ c3 such that ci ∈ [ai , bi ] and f (ci ) =

, i = 1, 2, 3?

bi − ai

(V. Brayman)

8. Call by Z-ball the set of points of the form

S = {(x, y, z)|x2 + y 2 + z 2 ≤ R2 , x, y, z, ∈ Z}, R ∈ R.

Prove that there exists no Z-ball which contains exactly 2005 distinct points.

(A. Bondarenko)

9. Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not

passing through the beginning of coordinates O. Call such triangle positive if for at least

−→

two of i = 1, 2, 3 vector OA turns counterclockwise when point A moves from Ai to Ai+1

(here A4 = A1 ) and negative otherwise. Let (xi , yi ) be coordinates of points Ai , i = 1, 2, 3.

Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for

positive triangles A1 A2 A3 and negative for negative ones.

(V. Grinberg, USA)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1. Let K be compact set in the space C([0, 1]) with uniform metric. Prove that the

function f (t) = min{x(t) + x(1 − t) : x ∈ K}, t ∈ [0, 1] is continuous.

(A. Kukush)

2. Find all λ ∈ C such that every sequence {an , n ≥ 1} ⊂ C which satisfy |λan+1 −λ2 an | < 1

for every n ≥ 1 is bounded.

(A. Prymak)

3. Let X and Y be linear normed spaces. Operator K : X → Y is said to be supercompact

if for every bounded set M ⊂ X the set K(M ) = {y ∈ Y | ∃x ∈ M : y = K(x)} is compact

in Y . Prove that the unique linear continuous supercompact operator from X to Y is zero

operator.

(I. Sinko)

2

4. Let A be real orthogonal matrix such that A = E. Prove that there exist orthogonal

matrix U and diagonal matrix B with entries ±1 at diagonal such that A = U BU T .

(A. Kukush)

5. Let B be bounded subset of connected metric space X. Prove or disprove that there

exist connected and bounded subset A ⊂ X such that B ⊂ A.

(M. Pupashenko)

+

6. Let t > 0, and let µ be a measure on Borel σ-field of R such that for every α < 1

exp(αxt ) dµ(x) < ∞. Prove that for every α < 1

R+

exp(α(x + 1)t ) dµ(x) < ∞.

R+

(A. Kukush)

7. Call by Z-ball the set of points of the form

S = {(x, y, z)|x2 + y 2 + z 2 ≤ R2 , x, y, z, ∈ Z}, R ∈ R.

Prove that there exists no Z-ball which contains exactly 2005 distinct points.

(A. Bondarenko)

8. Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not

passing through the beginning of coordinates O. Call such triangle positive if for at least

−→

two of i = 1, 2, 3 vector OA turns counterclockwise when point A moves from Ai to Ai+1

(here A4 = A1 ) and negative otherwise. Let (xi , yi ) be coordinates of points Ai , i = 1, 2, 3.

Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for

(V. Grinberg, USA)

positive triangles A1 A2 A3 and negative for negative ones.

9. Let x0 < x1 < · · · < xn and y0 < y1 < · · · < yn . Prove that there exists a strictly

increasing on [x0 , xn ] polynomial p such that p(xj ) = yj , j = 0, . . . , n.

(A. Prymak)

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 1-2 years students.

1. Is it true that the sequence {xn , n ≥ 1} of real numbers is convergent if and only if

lim lim |xn − xm | = 0?

(O. Nesterenko)

n→∞ m→∞

2. Let A, B, C be real matrices. Prove the inequality

tr(A(AT − B T ) + B(B T − C T ) + C(C T − AT )) ≥ 0.

(M. Vyazovska)

3. Billiard table is obtained by cutting some squares of a chessboard. Billiard ball is

shot from one of the table corners such that its trajectory forms angle α with the side of

the billiard table, tg α ∈ Q. When the ball hit the border of the billiard table it reflects

according to the rule: the angle of incidence equals angle of reflection. If the ball landing

in any corner it falls into a hole. Prove that the ball will necessarily fall into a hole.

(G. Kryukova)

4. Solve an equation lim

n→∞

1+

x+

x2 + · · · +

√

xn = 2.

(A. Kukush)

5. Do there exist matrices A, B, C which have no common eigenvectors and satisfy the

condition AB = BC = CA?

(V. Brayman)

π

6. Prove that

7. Let f ∈ C

cos 2x cos 3x cos 4x . . . cos 2005x dx > 0.

−π

(1)

(M. Pupashenko)

(R) and a1 < a2 < a3 < b1 < b2 < b3 . Prove or disprove that there exist

f (bi ) − f (ai )

real numbers c1 ≤ c2 ≤ c3 such that ci ∈ [ai , bi ] and f (ci ) =

, i = 1, 2, 3?

bi − ai

(V. Brayman)

8. Call by Z-ball the set of points of the form

S = {(x, y, z)|x2 + y 2 + z 2 ≤ R2 , x, y, z, ∈ Z}, R ∈ R.

Prove that there exists no Z-ball which contains exactly 2005 distinct points.

(A. Bondarenko)

9. Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not

passing through the beginning of coordinates O. Call such triangle positive if for at least

−→

two of i = 1, 2, 3 vector OA turns counterclockwise when point A moves from Ai to Ai+1

(here A4 = A1 ) and negative otherwise. Let (xi , yi ) be coordinates of points Ai , i = 1, 2, 3.

Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for

positive triangles A1 A2 A3 and negative for negative ones.

(V. Grinberg, USA)

Open Competition for University Students

of Mechanics and Mathematics Faculty

of Kyiv State Taras Shevchenko University.

Problems for 3-4 years students.

1. Let K be compact set in the space C([0, 1]) with uniform metric. Prove that the

function f (t) = min{x(t) + x(1 − t) : x ∈ K}, t ∈ [0, 1] is continuous.

(A. Kukush)

2. Find all λ ∈ C such that every sequence {an , n ≥ 1} ⊂ C which satisfy |λan+1 −λ2 an | < 1

for every n ≥ 1 is bounded.

(A. Prymak)

3. Let X and Y be linear normed spaces. Operator K : X → Y is said to be supercompact

if for every bounded set M ⊂ X the set K(M ) = {y ∈ Y | ∃x ∈ M : y = K(x)} is compact

in Y . Prove that the unique linear continuous supercompact operator from X to Y is zero

operator.

(I. Sinko)

2

4. Let A be real orthogonal matrix such that A = E. Prove that there exist orthogonal

matrix U and diagonal matrix B with entries ±1 at diagonal such that A = U BU T .

(A. Kukush)

5. Let B be bounded subset of connected metric space X. Prove or disprove that there

exist connected and bounded subset A ⊂ X such that B ⊂ A.

(M. Pupashenko)

+

6. Let t > 0, and let µ be a measure on Borel σ-field of R such that for every α < 1

exp(αxt ) dµ(x) < ∞. Prove that for every α < 1

R+

exp(α(x + 1)t ) dµ(x) < ∞.

R+

(A. Kukush)

7. Call by Z-ball the set of points of the form

S = {(x, y, z)|x2 + y 2 + z 2 ≤ R2 , x, y, z, ∈ Z}, R ∈ R.

Prove that there exists no Z-ball which contains exactly 2005 distinct points.

(A. Bondarenko)

8. Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not

passing through the beginning of coordinates O. Call such triangle positive if for at least

−→

two of i = 1, 2, 3 vector OA turns counterclockwise when point A moves from Ai to Ai+1

(here A4 = A1 ) and negative otherwise. Let (xi , yi ) be coordinates of points Ai , i = 1, 2, 3.

Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for

(V. Grinberg, USA)

positive triangles A1 A2 A3 and negative for negative ones.

9. Let x0 < x1 < · · · < xn and y0 < y1 < · · · < yn . Prove that there exists a strictly

increasing on [x0 , xn ] polynomial p such that p(xj ) = yj , j = 0, . . . , n.

(A. Prymak)