Seven students are standing on a straight line, one after the other. Three of the students, let’s call them \(A,B,\) and \(C\) behave badly and can’t be next to each other. For example: \(\star \star AB\star \star C\) and \(\star ABC\star \star \star\) are invalid arrangements, where the star denotes any other student. However, \(A\star B\star \star \star C\) is an example of a valid arrangement. How many valid arrangements are there?
A point \(P\) and a line \(L\) are drawn on a piece of paper. What is the shortest path from \(P\) to \(L\)? You should give a proof that your path is indeed the shortest.
Show that if any \(12\) two-digit numbers are given, you can always choose two of them such that their difference is of the form \(AA\) where \(A\) is some digit from \(0\) to \(9\).
Let \(a,b,c\) be whole numbers. Show that if \(a^2+b^2=c^2\), then at least one of \(a,b,\) or \(c\) must be even.