Open Competition for University Students of The Faculty of Mechanics and Mathematics of Kyiv National Taras Shevchenko University Problems for 1-2 years students n (k + p)(k + q) n→∞ (k + r)(k + s) k=1 Let p, q, r, s be positive integers Find the limit lim n Is it true that for every n k(2n k ) is divisible by 8? the number k=1 Two players in turn replace asterisks in the matrix ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (R Ushakov) (A Kukush) of size 10 × 10 by positive integers 1, , 100 (at each turn one may take any number which has not been used earlier and replace by it any asterisk) If they form a non-singular matrix then the first player wins, else the second player wins Has anybody of players a winning strategy? If somebody has, then who? (V Brayman) Prove that a function f ∈ C (0, +∞) which satisfy f (x) = , x > 0, + x + cos f (x) is bounded at (0, +∞) (O Nesterenko) Does there exist a polynomial, which takes value k exactly at k distinct real points for every k 2007? (V Brayman) The clock-face is a dice of radius The hour-hand is a dice of radius 1/2 touching the circle of the clock-face in inner way, and the minute-hand is a segment of length Find the area of the figure formed by all intersections of hands in 12 hours (i.e in one full turn of the hour-hand) (G Shevchenko) Find the maximum of x31 + + x310 for x1 , , x10 ∈ [−1, 2] such that x1 + + x10 = 10 (D Mitin) Let a0 = 1, a1 = and an = an−1 + (n − 1)an−2 , n Prove that for every odd number p the number ap − is divisible by p (O Rybak) Find all positive integers n for which there exist infinitely many matrices A of size n × n with integer entries such that An = I (here I is the identity matrix) (A Bondarenko, M Vyazovska) Open Competition for University Students of The Faculty of Mechanics and Mathematics of Kyiv National Taras Shevchenko University Problems for 3-4 years students ∞ Does the Riemann integral sin x dx converge? x + ln x (A Kukush) The clock-face is a dice of radius The hour-hand is a dice of radius 1/2 touching the circle of the clock-face in inner way, and the minute-hand is a segment of length Find the area of the figure formed by all intersections of hands in 12 hours (i.e in one full turn of the hour-hand) (G Shevchenko) Prove that a function f ∈ C (0, +∞) which satisfy , x > 0, f (x) = + x + cos f (x) is bounded at (0, +∞) (O Nesterenko) Does there exist a polynomial, which takes value k exactly at k distinct real points for every k 2007? (V Brayman) Let f : R → [0, +∞) be measurable function such that A f dλ < +∞ for every set A of finite Lebesgue measure (i.e λ(A) < +∞) Prove that there exist a constant M and Lebesgue integrable function g : R → [0, +∞) such that f (x) g(x) + M, x ∈ R (V Radchenko) Investigate the character of monotonicity of a function f (σ) = E , σ > 0, where ξ is a + eξ gaussian random variable with mean m and covariance σ (m is a real parameter) (A Kukush) 3 Find the maximum of x1 + + x10 for x1 , , x10 ∈ [−1, 2] such that x1 + + x10 = 10 (D Mitin) Let A, B be symmetric real positively defined matrices and the matrix A + B − E is positively defined as well Is it possible that the matrix A−1 +B −1 − 21 (A−1 B −1 +B −1 A−1 ) is negatively defined? (A Kukush) Let P (z) be polynomial with leading coefficient Prose that there exists a point z0 at the unit circle {z ∈ C : |z| = 1} such that |P (z0 )| (O Rybak)