Problems(2)
Inequality with condition a + b + c = 1
Source: Austrian Mathematical Olympiad 2008 Problem 1
2/6/2009
Prove the inequality
\sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3}
holds for all positive real numbers , and with a \plus{} b \plus{} c \equal{} 1.
inequalitiesfunctionlogarithmsinequalities unsolved
Determine all functions
Source: Austrian Mathematical Olympiad 2008 Problem 4
2/6/2009
Determine all functions mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1) f(mn) \equal{} f(m)\plus{}f(n),
(2) f(2008) \equal{} 0, and
(3) f(n) \equal{} 0 for all .
functionmodular arithmeticalgebra unsolvedalgebra