Two boxes together contain 65 balls of various sizes. Each ball is white, black, red, or yellow. Each time we take five balls of the same color, at least two are of the same size.

(a) What is the maximum number of types of balls that exist in the boxes? Two balls are considered of different types when they have different colors or sizes.

(b) Show that there are at least three balls, which are in the same box, and which are of the same type.

Hint: Observe that for a token to be removed it had to be flipped an odd number of times, and all circles have an even number of neighbors.
Hint: There are at most 4 distinct sizes of balls for each color.

Hint: Being a/600 and b/700 the two fractions, check which factors the numerator and the denominator of the sum can have in common.

Facts that Help: A fraction is said to be irreducible if the numerator and the denominator do not have common prime factors.
Hint: Observe that the particular values of \(x_{1}, x_{2}, \ldots, x_{n}\) are not important, but rather the number of these that are equal to 1 and -2.
(a) There cannot be five balls of the same color and different sizes because every time we pick five balls of the same color, two must be of the same size. Thus, there are at most four sizes for each color. Therefore, there are at most \(4 \times 4=16\) types of balls.

(b) The two boxes together contain 65 balls, and one of them must contain at least 33 balls. On the other hand, there are at most 16 types of balls, and since \(2 \times 16=32<33\), we conclude that this box contains three or more balls of the same type.