Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems for 1-2 years students Does there exist a function F : R2 → N such that the equality F (x, y) = F (y, z) holds if and only if x = y = z? (A Bondarenko, A Prymak) sin x cos x Consider graphs of functions y = a +a , where a ∈ [1; 2,5] Prove that there exists a point M such that the distance from M to any of these graphs is less then 0,4 (A Kukush) (1) Consider a function f ∈ C ([−1, 1]) such that f (−1) = f (1) = Prove that ∃ x ∈ [−1, 1] : f (x) = (1 + x2 )f (x) (A Prymak) Each entry of the matrix A = (aij ) ∈ Mn (R) is equal to or and moreover aii = 0, aij + aji = (1 ≤ i < j ≤ n) Prove that rkA n − (A Oliynyk) π Prove the inequality (cos x)sin x dx < (cos x)sin x + (sin x)cos x (A Kukush) Find the dimension of the subspace of those linear operators ϕ on Mn (R) for which the identity ϕ(AT ) = (ϕ(A))T holds for every matrix A ∈ Mn (R) (A Oliynyk) ∞ k j For every k ∈ N prove that ak = ∈ / Q j! j=1 (V Brayman, Yu Shelyazhenko) Find all functions f ∈ C(R) such that ∀ x, y, z ∈ R holds f (x) + f (y) + f (z) = f 37 x + 67 y − 27 z + f 67 x − 27 y + 37 z + f − 72 x + 37 y + 76 z (V Brayman) Construct a set A ⊂ R and a function f : A → R such that ∀ a1 , a2 ∈ A |f (a1 ) − f (a2 )| ≤ |a1 − a2 |3 and the range of f is uncountable (V Brayman) 10 Prizmatoid is a convex polyhedron all the vertices of which lie in two parallel planes – the lower and the upper bases of prizmatoid Consider a section of a given prizmatoid by a plane which is parallel to the bases and is at distance x from the lower base Prove that the area of this section is a polynomial of x of at most second degree (A Kukush, R Ushakov) Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems for 3-4 years students Let ξ be a random variable with finite expectation at probability space (Ω, F, P ) Let ω be a signed measure on F such that ∀ A ∈ F : inf ξ(x) · P (A) ≤ ω(A) ≤ sup ξ(x) · P (A) x∈A Prove that ∀ A ∈ F : ω(A) = x∈A ξ(x)dP (x) (A Kukush) A For every positive integer n consider function fn (x) = nsin x + ncos x , x ∈ R Prove that there exists a sequence {xn } such that for every n fn has a global maximum at xn and xn → as n → ∞ (A Kukush) Let U be nonsingular real n × n matrix a ∈ Rn and let L be the subspace of Rn Prove that PU T L (U −1 a) ≤ U −1 · PL a , where PM is a projector onto subspace M (A Kukush) Let f : C\{0} → (0, +∞) be continuous function such that lim f (z) = 0, lim f (z) = z→0 |z|→∞ dz ∞ Prove that for every T > there exist a solution of differential equation = izf (z) dt which has period T (O Stanzhitskyy) π Prove the inequality (cos x)sin x dx < (cos x)sin x + (sin x)cos x (A Kukush) Find the dimension of the subspace of those linear operators ϕ on Mn (R) for which the identity ϕ(AT ) = (ϕ(A))T holds for every matrix A ∈ Mn (R) (A Oliynyk) ∞ k j For every k ∈ N prove that ak = ∈ / Q j! j=1 (V Brayman, Yu Shelyazhenko) Find all functions f ∈ C(R) such that ∀ x, y, z ∈ R holds f (x) + f (y) + f (z) = f 37 x + 67 y − 27 z + f 67 x − 27 y + 37 z + f − 72 x + 37 y + 76 z (V Brayman) Construct a set A ⊂ R and a function f : A → R such that ∀ a1 , a2 ∈ A |f (a1 ) − f (a2 )| ≤ |a1 − a2 |3 and the range of f is uncountable (V Brayman) 10 Prizmatoid is a convex polyhedron all the vertices of which lie in two parallel planes – the lower and the upper bases of prizmatoid Consider a section of a given prizmatoid by a plane which is parallel to the bases and is at distance x from the lower base Prove that the area of this section is a polynomial of x of at most second degree (A Kukush, R Ushakov)