mechmat competition2006

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Open Competition for University Students of The Faculty of Mechanics and Mathematics of Kyiv National Taras Shevchenko University Problems for 1-2 years students Find all positive integers n such that the polinomial (x4 − 1)n + (x2 − x)n is divisible by x5 − (A Kukush) Let z ∈ C be such that points z , 2z + z , 3z + 3z + z and 4z + 6z + 4z + are the vertices of an inscribed quadrangle at complex plane Find Re z (V Brayman) n Find the minimum over all unit vectors x1 , , xn+1 ∈ R of max (xi , xj ) i holds.) (A Kukush) Does there exist an infinite set of square symmetric matrices M such that for any distinct matrices A, B ∈ M AB = B A holds but AB = BA? (V Brayman) f (x) Let f : (0, +∞) → R be continuous concave function, lim f (x) = +∞, lim = Prove x→+∞ x→+∞ x that sup {f (n)} = 1, where {a} = a − [a] is fractional part of a n∈N (O Nesterenko) Does there exist a continuous function f : R → (0, 1) such that the sequence n n f (x) dx, n an = −n 1, converges and the sequence bn = f (x) ln f (x) dx, n 1, is divergent? −n (A Kukush) Let P (x) be a polinomial such that there exist infinitely many pairs of integers (a, b) such that P (a + 3b) + P (5a + 7b) = Prove that the polinomial P (x) has an integer root (V Brayman) n aj For every real numbers a1 , a2 , , an ∈ R \ {0} prove the inequality a + a2j i,j=1 i (S Novak, Great Britain) Open Competition for University Students of The Faculty of Mechanics and Mathematics of Kyiv National Taras Shevchenko University Problems for 3-4 years students Let z ∈ C be such that points z , 2z + z , 3z + 3z + z and 4z + 6z + 4z + are the vertices of an inscribed quadrangle at complex plane Find Re z (V Brayman) ∞ Does there exist a continuous function f : R → (0, 1) such that ∞ while f (x) dx < ∞ −∞ f (x) ln f (x) dx is divergent? −∞ m×n (A Kukush) and let B be symmetric n × n matrix such that Let Em be m × m identity matrix, A ∈ R Em A block matrix is positively defined Prove that “matrix determinant” B − AT A is also AT B positively defined (A Kukush) Does there exist an infinite set of square symmetric matrices M such that for any distinct matrices A, B ∈ M holds AB = B A but AB = BA? (V Brayman) a) Let ξ and η be random variables (not necessarily independent) which have continuous distribution functions Prove that min(ξ, η) also has continuous distribution function b) Let ξ and η be random variables which have densities Is it true that min(ξ, η) also has a density? (A Kukush, G Shevchenko) Is it possible to choose uncountable set A ⊂ l2 of elements with unit norm such that for any ∞ distinct x = (x1 , , xn , ), y = (y1 , , yn , ) from A the series |xn − yn | is divergent? n=1 (A Bondarenko) Let ξ, η be independent identically distributed random variables such that ξη P (ξ = 0) = Prove the inequality E ξ + η2 (S Novak, Great Britain) For every n ∈ N find the minimal λ > such that for every convex compact set K ⊂ Rn there exist a point x ∈ K such that the set which is homothetic to K with centre x and coefficient (−λ) contains K (O Lytvak, Canada) Let X = L1 [0, 1] and let Tn : X → X be the sequence of nonnegative (i.e f =⇒ Tn f 0) linear operators such that Tn and lim f − Tn f X = for f (x) ≡ x and for f (x) ≡ Prove that lim f − Tn f n→∞ n→∞ X = for every f ∈ X (A Prymak)
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