BRATISLAVA MATHEMATICS CORRESPONDENCE SEMINAR
Faculty of Mathematics and Physics of Comenius University
Union of Slovak Mathematicians and Physicists
Centre of Spare Time - IUVENTA

BMCS - Problem set of the 3rd spring series 1997/98

Sequences and sums

  1. Consider the following sequence: a1=4, a2=9, a(n+2)=a(n+1) + an + 1, for all positive integers n. Can you find positive integers k, l, m, such that ak . al = am ?

  2. Let a1, a2, ..., an are different positive integers (all pairs are different). Prove that
    SUM(for k=1..n) of (ak/(k.k)) >= SUM(for k=1..n) of (1/k).

  3. Find the sum S = arctg(1/2) + arctg(1/8) + ... + arctg(1/(2.n.n)).

  4. Let all coefficients of the polynom P(x) are integers. Examine the sequence:
    a0=0, an = P(a(n-1)), for all integers n
    and prove the following for all integers k, m:
    a(k,m) = (ak,am)
    where (p,q) is the gratest common divisor of p, q.

  5. Having a positive real number z, we can construct a sequence:
    P(z) = { [z], [2z], [3z], ... }
    where [a] is the floor integer part of a real number a. Find the necessary and sufficient conditions for real numbers x,y of factorizing the set of all positive integers by the sequences P(x) and P(y)

    Note: Sets A, B factorize the set C, if the union of A and B is C and the intersection of A and B is the empty set.