Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems for 1-2 years students Is it true that the sequence {xn , n ≥ 1} of real numbers is convergent if and only if lim lim |xn − xm | = 0? (O Nesterenko) n→∞ m→∞ Let A, B, C be real matrices Prove the inequality tr(A(AT − B T ) + B(B T − C T ) + C(C T − AT )) ≥ (M Vyazovska) Billiard table is obtained by cutting some squares of a chessboard Billiard ball is shot from one of the table corners such that its trajectory forms angle α with the side of the billiard table, tg α ∈ Q When the ball hit the border of the billiard table it reflects according to the rule: the angle of incidence equals angle of reflection If the ball landing in any corner it falls into a hole Prove that the ball will necessarily fall into a hole (G Kryukova) Solve an equation lim n→∞ 1+ x+ x2 + · · · + √ xn = (A Kukush) Do there exist matrices A, B, C which have no common eigenvectors and satisfy the condition AB = BC = CA? (V Brayman) π Prove that Let f ∈ C cos 2x cos 3x cos 4x cos 2005x dx > −π (1) (M Pupashenko) (R) and a1 < a2 < a3 < b1 < b2 < b3 Prove or disprove that there exist f (bi ) − f (ai ) real numbers c1 ≤ c2 ≤ c3 such that ci ∈ [ai , bi ] and f (ci ) = , i = 1, 2, 3? bi − (V Brayman) Call by Z-ball the set of points of the form S = {(x, y, z)|x2 + y + z ≤ R2 , x, y, z, ∈ Z}, R ∈ R Prove that there exists no Z-ball which contains exactly 2005 distinct points (A Bondarenko) Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not passing through the beginning of coordinates O Call such triangle positive if for at least −→ two of i = 1, 2, vector OA turns counterclockwise when point A moves from Ai to Ai+1 (here A4 = A1 ) and negative otherwise Let (xi , yi ) be coordinates of points Ai , i = 1, 2, Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for positive triangles A1 A2 A3 and negative for negative ones (V Grinberg, USA) Open Competition for University Students of Mechanics and Mathematics Faculty of Kyiv State Taras Shevchenko University Problems for 3-4 years students Let K be compact set in the space C([0, 1]) with uniform metric Prove that the function f (t) = min{x(t) + x(1 − t) : x ∈ K}, t ∈ [0, 1] is continuous (A Kukush) Find all λ ∈ C such that every sequence {an , n ≥ 1} ⊂ C which satisfy |λan+1 −λ2 an | < for every n ≥ is bounded (A Prymak) Let X and Y be linear normed spaces Operator K : X → Y is said to be supercompact if for every bounded set M ⊂ X the set K(M ) = {y ∈ Y | ∃x ∈ M : y = K(x)} is compact in Y Prove that the unique linear continuous supercompact operator from X to Y is zero operator (I Sinko) Let A be real orthogonal matrix such that A = E Prove that there exist orthogonal matrix U and diagonal matrix B with entries ±1 at diagonal such that A = U BU T (A Kukush) Let B be bounded subset of connected metric space X Prove or disprove that there exist connected and bounded subset A ⊂ X such that B ⊂ A (M Pupashenko) + Let t > 0, and let µ be a measure on Borel σ-field of R such that for every α < exp(αxt ) dµ(x) < ∞ Prove that for every α < R+ exp(α(x + 1)t ) dµ(x) < ∞ R+ (A Kukush) Call by Z-ball the set of points of the form S = {(x, y, z)|x2 + y + z ≤ R2 , x, y, z, ∈ Z}, R ∈ R Prove that there exists no Z-ball which contains exactly 2005 distinct points (A Bondarenko) Consider triangles A1 A2 A3 at cartesian plane with sides and their extensions not passing through the beginning of coordinates O Call such triangle positive if for at least −→ two of i = 1, 2, vector OA turns counterclockwise when point A moves from Ai to Ai+1 (here A4 = A1 ) and negative otherwise Let (xi , yi ) be coordinates of points Ai , i = 1, 2, Prove that there exists no polynomial P (x1 , y1 , x2 , y2 , x3 , y3 ) which values are positive for (V Grinberg, USA) positive triangles A1 A2 A3 and negative for negative ones Let x0 < x1 < · · · < xn and y0 < y1 < · · · < yn Prove that there exists a strictly increasing on [x0 , xn ] polynomial p such that p(xj ) = yj , j = 0, , n (A Prymak)