A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?
A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.
Every year the citizens of the planet “Lotsofteeth" enter a contest
to see who has the most teeth.
This year the judge notices:
Nobody has 0 teeth (everyone has at least 1).
There are more people in the contest than the most teeth that any one person has. (For example, if the most teeth anyone has is 27, then there are more than 27 people participating in the contest.)
Must there be two people who have exactly the same number of teeth? Explain why.