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 2nd spring series 1996/97

Inequalities

  1. Solve the equation in the domain of positive reals:
    (x4+y2)(z2+x2)(16y2+x2)=32x4y2z

  2. Prove that the formula
    x12+x22 +x32+x42+x52 >=(2 /3)*(x1x2+x2x3+x3x4+x4x5)
    holds for all tuplets of 5 reals x1, x2, x3, x4, x5.
    .

  3. Prove that for all integers n the following is true:
    (1+(1 / n))n < (1+1 / (n+1))n+1.

  4. Prove that
    (a5+b5+c5) / (a2b2c2) >= 1/a + 1/b + 1/c
    for arbitrary positive reals a, b, c.

  5. Let a,b,c are positive real numbers. Then
    c.(a2-ab+b2)+a.(b2-bc+c2) >= b.(a2+ac+c2).
    Prove.