A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, in such a way that the distance between their centers is 60 mm. Calculate the radius of the hole.

Emilia and Julieta have a pile of 2024 cards and play the following game: they take turns, and each player removes a number of cards that must be a power of two, i.e., \(1, 2, 4, 8, \dots\). The player who removes the last card wins. Julieta starts the game. Prove that there exists a strategy for Julieta that guarantees her victory, no matter how Emilia plays.

Determine all triples \( (a, b, c) \) of positive integers such that: \[ a + b + c = abc. \]

Consider a triangle with sides of length \( a \), \( b \), and \( c \) that satisfy the following conditions: \[ a + b = c + 3 \quad c^2 + 9 = 2ab \] Find the area of the triangle.

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red. 1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \). 2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).