Open Competition for University Students

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 1-2 years students

1. Triangle ABC is inscribed into a circle. Does there exist a point D on this circle such that ABCD

is a circumscribed quadrilateral?

(A. Kukush)

2. Let F0 = 0, F1 = 1, Fk = Fk−1 + Fk−2 , k 2 be a Fibonacci sequence. Find all positive integers n

for which the polynomial Fn xn+1 + Fn+1 xn − 1 is irreducible in the ring of polynomials with rational

coefficients Q[x].

(R. Ushakov)

3. Let A, B, C be angles of acute triangle. Prove the inequalities:

a)

cos A

cos B

cos C

+

+

sin B sin C sin C sin A sin A sin B

b) √

2;

cos A

cos B

cos C

+√

+√

sin B sin C

sin C sin A

sin A sin B

√

3.

(A. Kukush, M. Rozhkova)

4. Find all positive integers n for which there exist matrices A, B, C ∈ Mn (Z) such that

ABC + BCA + CAB = I.

Here I is the identity matrix.

(V. Brayman)

5. Let x, y : R → R be a pair of functions such that x(t) − x(s) y(t) − y(s)

0 for all t, s ∈ R.

Prove that there exist two non-decreasing functions f, g : R → R and function z : R → R such that

x(t) = f (z(t)) and y(t) = g(z(t)) for all t ∈ R.

(J. Dhaene (Belgium), V. Brayman)

6. Let {xn , n

n

1

xk .

n→∞ n k=1

1} be a sequence of real numbers such that there exists finite limit lim

n

1

k p−1 xk .

p

n→∞ n k=1

Prove that for every p > 1 there exists finite limit lim

7. Let K(x) = xe−x , x ∈ R. For every n

sup

(D. Mitin)

3 determine

min K(|xi − xj |).

x1 ,...,xn ∈R 1 i

(A. Bondarenko, E. Saff (USA))

8. Does there exist a function f : Q → Q such that f (x)f (y) |x − y| holds for any x, y ∈ Q, x = y,

and for each x ∈ Q the set y ∈ Q | f (x)f (y) = |x − y| is infinite?

(V. Brayman)

9. Find all n 2 for which it is possible to enumerate all permutations of set {1, . . . , n} with numbers

1, . . . , n! in such way that for any pair of permutations σ, τ with adjacent numbers, as well as for

pair with numbers 1 and n!, σ(k) = τ (k) holds for every 1 k n.

(O. Rudenko)

Open Competition for University Students

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 3-4 years students

1. Find all positive integers n for which there exist matrices A, B, C ∈ Mn (Z) such that

ABC + BCA + CAB = I.

Here I is the identity matrix.

2. Let {xn , n

(V. Brayman)

n

1

xk .

n→∞ n k=1

1} be a sequence of real numbers such that there exists finite limit lim

n

1

k p−1 xk .

p

n→∞ n k=1

Prove that for every p > 1 there exists finite limit lim

(D. Mitin)

3. Let x, y : R → R be a pair of functions such that x(t) − x(s) y(t) − y(s)

0 for all t, s ∈ R.

Prove that there exist two non-decreasing functions f, g : R → R and function z : R → R such that

x(t) = f (z(t)) and y(t) = g(z(t)) for all t ∈ R.

(J. Dhaene (Belgium), V. Brayman)

4. Let K(x) = xe−x , x ∈ R. For every n 3 determine

sup

min K(|xi − xj |).

x1 ,...,xn ∈R 1 i

(A. Bondarenko, E. Saff (USA))

5. Does there exist a function f : Q → Q such that f (x)f (y) |x − y| holds for any x, y ∈ Q, x = y,

and for each x ∈ Q the set y ∈ Q | f (x)f (y) = |x − y| is infinite?

(V. Brayman)

ax

6. Let µ be a measure on Borel σ-algebra in R, such that R e dµ(x) < ∞ for all a ∈ R, and

µ (−∞, 0) > 0, µ (0, +∞) > 0. Prove that there exists unique real a such that R xeax dµ(x) = 0.

(A. Kukush)

7. Let {ξn , n 0} and {νn , n 1} be two independent sets of i.i.d. random variables (distribution

may be different in each set). It is known that Eξ0 = 0 and P {ν1 = 1} = p, P {ν1 = 0} = 1 − p,

n

1 n

p ∈ (0, 1). Denote x0 = 0 and xn =

νk , n 1. Prove that

ξx → 0, n → ∞, with probability

n k=0 k

k=1

one.

(A. Dorogovtsev)

8. Let X1 , . . . , X2n be a set of i.i.d. random variables such that X1 = 0 a.s. Denote

Yk =

k

i=1

Xi

k

i=1

, 1

k

2n.

Xi2

2

) 1 + 4(EYn )2 .

(S. Novak, Great Britain)

Prove the inequality E(Y2n

9. Find all n 2 for which it is possible to enumerate all permutations of set {1, . . . , n} with numbers

1, . . . , n! in such way that for any pair of permutations σ, τ with adjacent numbers, as well as for

pair with numbers 1 and n!, σ(k) = τ (k) holds for every 1 k n.

(O. Rudenko)

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 1-2 years students

1. Triangle ABC is inscribed into a circle. Does there exist a point D on this circle such that ABCD

is a circumscribed quadrilateral?

(A. Kukush)

2. Let F0 = 0, F1 = 1, Fk = Fk−1 + Fk−2 , k 2 be a Fibonacci sequence. Find all positive integers n

for which the polynomial Fn xn+1 + Fn+1 xn − 1 is irreducible in the ring of polynomials with rational

coefficients Q[x].

(R. Ushakov)

3. Let A, B, C be angles of acute triangle. Prove the inequalities:

a)

cos A

cos B

cos C

+

+

sin B sin C sin C sin A sin A sin B

b) √

2;

cos A

cos B

cos C

+√

+√

sin B sin C

sin C sin A

sin A sin B

√

3.

(A. Kukush, M. Rozhkova)

4. Find all positive integers n for which there exist matrices A, B, C ∈ Mn (Z) such that

ABC + BCA + CAB = I.

Here I is the identity matrix.

(V. Brayman)

5. Let x, y : R → R be a pair of functions such that x(t) − x(s) y(t) − y(s)

0 for all t, s ∈ R.

Prove that there exist two non-decreasing functions f, g : R → R and function z : R → R such that

x(t) = f (z(t)) and y(t) = g(z(t)) for all t ∈ R.

(J. Dhaene (Belgium), V. Brayman)

6. Let {xn , n

n

1

xk .

n→∞ n k=1

1} be a sequence of real numbers such that there exists finite limit lim

n

1

k p−1 xk .

p

n→∞ n k=1

Prove that for every p > 1 there exists finite limit lim

7. Let K(x) = xe−x , x ∈ R. For every n

sup

(D. Mitin)

3 determine

min K(|xi − xj |).

x1 ,...,xn ∈R 1 i

(A. Bondarenko, E. Saff (USA))

8. Does there exist a function f : Q → Q such that f (x)f (y) |x − y| holds for any x, y ∈ Q, x = y,

and for each x ∈ Q the set y ∈ Q | f (x)f (y) = |x − y| is infinite?

(V. Brayman)

9. Find all n 2 for which it is possible to enumerate all permutations of set {1, . . . , n} with numbers

1, . . . , n! in such way that for any pair of permutations σ, τ with adjacent numbers, as well as for

pair with numbers 1 and n!, σ(k) = τ (k) holds for every 1 k n.

(O. Rudenko)

Open Competition for University Students

of The Faculty of Mechanics and Mathematics

of Kyiv National Taras Shevchenko University

Problems for 3-4 years students

1. Find all positive integers n for which there exist matrices A, B, C ∈ Mn (Z) such that

ABC + BCA + CAB = I.

Here I is the identity matrix.

2. Let {xn , n

(V. Brayman)

n

1

xk .

n→∞ n k=1

1} be a sequence of real numbers such that there exists finite limit lim

n

1

k p−1 xk .

p

n→∞ n k=1

Prove that for every p > 1 there exists finite limit lim

(D. Mitin)

3. Let x, y : R → R be a pair of functions such that x(t) − x(s) y(t) − y(s)

0 for all t, s ∈ R.

Prove that there exist two non-decreasing functions f, g : R → R and function z : R → R such that

x(t) = f (z(t)) and y(t) = g(z(t)) for all t ∈ R.

(J. Dhaene (Belgium), V. Brayman)

4. Let K(x) = xe−x , x ∈ R. For every n 3 determine

sup

min K(|xi − xj |).

x1 ,...,xn ∈R 1 i

(A. Bondarenko, E. Saff (USA))

5. Does there exist a function f : Q → Q such that f (x)f (y) |x − y| holds for any x, y ∈ Q, x = y,

and for each x ∈ Q the set y ∈ Q | f (x)f (y) = |x − y| is infinite?

(V. Brayman)

ax

6. Let µ be a measure on Borel σ-algebra in R, such that R e dµ(x) < ∞ for all a ∈ R, and

µ (−∞, 0) > 0, µ (0, +∞) > 0. Prove that there exists unique real a such that R xeax dµ(x) = 0.

(A. Kukush)

7. Let {ξn , n 0} and {νn , n 1} be two independent sets of i.i.d. random variables (distribution

may be different in each set). It is known that Eξ0 = 0 and P {ν1 = 1} = p, P {ν1 = 0} = 1 − p,

n

1 n

p ∈ (0, 1). Denote x0 = 0 and xn =

νk , n 1. Prove that

ξx → 0, n → ∞, with probability

n k=0 k

k=1

one.

(A. Dorogovtsev)

8. Let X1 , . . . , X2n be a set of i.i.d. random variables such that X1 = 0 a.s. Denote

Yk =

k

i=1

Xi

k

i=1

, 1

k

2n.

Xi2

2

) 1 + 4(EYn )2 .

(S. Novak, Great Britain)

Prove the inequality E(Y2n

9. Find all n 2 for which it is possible to enumerate all permutations of set {1, . . . , n} with numbers

1, . . . , n! in such way that for any pair of permutations σ, τ with adjacent numbers, as well as for

pair with numbers 1 and n!, σ(k) = τ (k) holds for every 1 k n.

(O. Rudenko)