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mechmat competition2009

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)



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