Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems for 1-2 years students Prove or disprove that lim |n sin n| = +∞ n→∞ Let f ∈ C (R) a) Prove that there exists θ ∈ R such that f (θ)f (θ) + 2(f (θ))2 ≥ b) Prove that there exists function G : R → R such that (∀x ∈ R f (x)f (x) + 2(f (x))2 ≥ 0)⇐⇒ G(f (x)) is convex on R √ √ n converges to some number a ∈ ( 32 e, 32 e) Prove that the sequence an = 32 · 54 · 98 · .· 2+1 n Find all complex solutions of the system of equations xk1 + xk2 + + xkn = 0, k = 1, 2, , n B ) then D = CA−1 B Let A be nonsingular matrix Prove that if rankA = rank ( CA D Let b(n, k) denotes the number of permutations of n elements in which just k elements remains at their places Calculate n b(n, k) k=1 Problems for 3-4 years students Solve all complex solutions of the system of equations xk1 + xk2 + + xkn = 0, k = 1, 2, , n Let A(t) be n × n matrix which is continuous on [0, +∞) Let B ⊂ Rn be the set of initial values x(0) such that the solution x(t) of dx = A(t)x is bounded on [0, +∞) Prove dt n that B is a subspace of R and if for every f ∈ C([0, +∞), Rn ) the system dx = A(t)x + f (t) (∗) dt has bounded on [0, +∞) solution then for every f ∈ C([0, +∞), Rn ) there exists unique solution x(t) of (∗) which is bounded on [0, +∞) and satisfy x(0) ∈ B ⊥ (B ⊥ denotes an orthogonal completion of B.) Let σ be arbitrary permutation of the set 1, 2, , n chosen at random (The probability to choose each permutation is n!1 ) Find the expectation of the number of elements which places are preserved by permutation σ Find all functions analytical in C \ {0} such that the image of any circle with center belongs to some circle with center (Here circle is a line.) Cone in Rn is a set obtained by transition and rotation from the set {(x1 , , xn ) : x21 + + x2n−1 ≤ rx2n } for some r > Prove that if A is non=bounded and convex subspace of Rn which contains no cone then there exists two-dimensional subspace B ⊂ Rn such that projection of A to B contains no cone in R2 Let {γk , k ≥ 1} be independent standard gaussian random variables Prove that max1≤k≤n γk2 ln n P : → 2, n → ∞ n n k=1 γk