Let be a group wit 10 naturals numbers.
1. Is there a non-empty subset that the sum of element divides by 10?
2. Is there a non-empty subset that the sum of elements divides by 11?
Point 1. Hint: can you find three equidistant points on the plane?
Point 2. Hint: Suppose we found 6 points located like this:
Let be the color of and assume that the triple equals the triple (pointwise). Can you find the required rectangle?
Now, can you find 6 points with this property?