Problem:
Which of the following describes the largest subset of values of y within the closed interval [0,π] for which
sin(x+y)≤sin(x)+sin(y)
for every x between 0 and π, inclusive?
Answer Choices:
A. y=0
B. 0≤y≤4π
C. 0≤y≤2π
D. 0≤y≤43π
E. 0≤y≤π
Solution:
If 0≤x≤π and 0≤y≤π, then sin(x)≥0, sin(y)≥0,cos(x)≤1, and cos(y)≤1. Therefore
sin(x+y)=sin(x)⋅cos(y)+cos(x)⋅sin(y)≤sin(x)+sin(y)
The given inequality holds for all y such that 0≤y≤π.
The problems on this page are the property of the MAA's American Mathematics Competitions