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 fall series 1997/98

Functions

  1. For a function f: Z -> R the following statement holds:

    f(z)=z-10, for z>100
    f(z)=f(f(z+11)), for z<=100.

    Prove, that for all z<=100: f(z)=91.

  2. Prove that the function f(x)=|x| (absolute value of x) can be expressed as a sum of two functions, both of which have central-symmetric graphs (the centres of symmetries don't have to be the same).

  3. Suppose a1, a2, ..., an are positive real numbers and b1, b2, ..., bn is their permutation. Find the maximum value of the product
    (a1+1/b1)(a2+1/b2)...(an+1/bn).

  4. f: R->R is continuous and f(f(f(x)))=x for each real number x. Prove that f(x)=x for all x real.

  5. f: N -> N has the following feature:
    f(n+1) > f(f(n))
    for each integer n. Prove that
    f(n)=n
    for each integer n.