# Understand the problem

Let be a real-valued function on the plane such that for every square in the plane, Does it follow that for all points in the plane?

##### Source of the problem

Putnam 2009 A1

##### Topic

Geometry

##### Difficulty Level

Easy

##### Suggested Book

Mathematical Olympiad Challenges by Titu Andreescu

# Start with hints

Do you really need a hint? Try it first!

If the function is indeed identically 0, then you would need a geometrical argument to prove it. If not, you would need to find an example of a function which is not identically zero and satisfies the hypothesis of the question. In principle, you should start by playing around with some examples.

The answer to the question is

**YES.** Think about squares that have as a vertex.

Consider the squares in picture. Clearly, Adding, we get . As both and are squares, the second and the third terms are zero. Thus, .

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