An ice cream machine distributes ice cream randomly. There are 5 flavours in the machine and you would like to have any one available flavour at least 3 times. How many total samples do you need to obtain to ensure that?
Prove that among \(11\) different infinite decimal fractions, you can choose two fractions which coincide in an infinite number of digits.
A convex polygon on the plane contains at least \(m^2+1\) points with integer coordinates. Prove that it contains \(m+1\) points with integers coordinates that lie on the same line.
Suppose a football team scores at least one goal in each of the \(20\) consecutive games. If it scores a total of \(30\) goals in those \(20\) games, prove that in some sequence of consecutive games it scores exactly \(9\) goals total.