# 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, .

# Watch the video (Coming Soon)

# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.