Some problems from Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University 1996 Let a, b, c ∈ C Find lim |a + b + c | n n n 1/n n→∞ Let function f ∈ C([1, +∞)) be such that for every x ≥ there exists a limit Ax lim A→∞ ϕ(x) f (u)du =: ϕ(x), ϕ(2) = and function ϕ is continuous at point x = Find A x The function f ∈ C([0, +∞)) is such that f (x) f (u)du → 1, x → +∞ Prove that 1/3 , x → +∞ 3x n−1 (x − xk ) sin(2πxk ) k=0 k+1 , where supremum is taken over all possible parti4 Find sup n−1 λ (xk+1 − xk )2 f (x) ∼ k=0 tions of [0, 1] of the form λ = {0 = x0 < x1 < < xn−1 < xn = 1}, n ≥ Let D be bounded connected domain with boundary ∂D and let f (z), F (z) be functions = for every z ∈ ∂D Prove that analytical in D It is known that F (z) = and Im Ff (z) (z) functions F (z) and F (z) + f (z) have equal number of zeroes in D Let A be linear operator in finite-dimensional space such that A1996 + A998 + 1996I = Prove that A has basis which consists of eigenvectors Let A1 , A2 , , An+1 be n × n matrices Prove that there exist numbers α1 , α2 , , αn+1 not all of which are zeroes such that the matrix α1 A1 + + αn+1 An+1 is singular Let matrix A be such that trA = Prove that there exist positive integer n and matrices A1 , , An such that A = A1 + + An and A2i = 0, ≤ i ≤ n 1997 Let ≤ k < n be positive integers Consider all possible representation of n as a sum of two or more positive integer summands (Two representations with differ by order of summands are assumed to be distinct) Prove that the number k appears as a summand exactly (n − k + 3)2n−k−2 times in these representations Prove that the field Q(x) of rational functions contains two subfields F and K such that [Q(x) : F ] < ∞ and [Q(x) : K] < ∞ but [Q(x) : (F K)] = ∞ Let the matrix A ∈ Mn (C) has unique eigenvalue a Prove that A commutes only with polynomials of A if and only if rank(A − aI) = n − Let a ∈ Rm be a vector-column Calculate (1 − aT (I + aaT )a)−1 +∞ Let f be positive non-increasing function on [1, +∞) such that xf (x)dx < ∞ +∞ f (x) Prove that the integral dx is convergent | sin x|1− x n j Find lim + n2 n→∞ j j=1 Let I be the interval of length at least and f be twice differentiable function on I such that |f (x)| ≤ and |f (x)| ≤ 1, x ∈ I Prove that |f (x)| ≤ 2, x ∈ I