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Problem 3 (due on Monday, October 10)

a) Is there a function f:RR such that f(x)+f(y)2f(x+y2)+sin2(xy) for all x,yR?

b) Is there a function f:RR such that f(x)+f(y)2f(x+y2)+sin|xy| for all x,yR?

The answer to a) is positive, for example f(x)=4x2 has the property. The answer to b) is negative. We received only one solution, from Prof. Vladislaw Kargin, who solved a) and b) under additional assumption about f (essentially that f is continuous). For a detailed solution and some related material see the following link Solution.