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5

Part of 2015 Costa Rica - Final Round

Problems(2)

1/(a^3 + 3b)+1/(b^3 + 3a)<=1/2 OLCOMA Costa Rica 2015 p5 day2

Source:

9/30/2021
Let a,b∈R+a,b \in R^+ with ab=1ab = 1, prove that 1a3+3b+1b3+3a≤12.\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.
algebrainequalities
f(a) + f(b)<= f (a + b) <= f(a) + f(b) + 1 if kf(n) <= f (kn) <= kf(n)+ k- 1

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 C2 p5

9/23/2021
Let f:N+→N+f: N^+ \to N^+ be a function that satisfies that kf(n)≤f(kn)≤kf(n)+k−1,  ∀k,n∈N+kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+ Prove that f(a)+f(b)≤f(a+b)≤f(a)+f(b)+1,  ∀a,b∈N+f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+
algebrainequalitiesFunctional inequalityfunctional