01a. PEN K12 (Canada 1969) Find all functions such that for all : , , .
01b. Let be a function satisfying the conditions:
(a) for all relatively prime and , and
(b) for all positive integers .
Show that there is a constant such that for all .
 P. Erdos, On the distribution function of additive functions, Ann. of Math., 47(1946), 1-20
 E. Howe, A new proof of Erdos’s theorem on monotone multiplicative functions, Amer. Math. Monthly, 93(1986), 593-595
 L. Moser and J. Lambek, On monotone multiplicative functions, Proc. Amer. Math. Soc., 4(1953), 544-545
Acknowledgement. We would like to express our gratitude to Andrei Frimu who helped us preparing the solutions and Orlando Doehring who offered me the solution file of CMO 2003.