Show that given \(11\) whole numbers, there will be at least \(2\) numbers whose difference ends in a zero.
Three whole numbers are marked on a number line. Show that for two of these marked numbers, the point halfway between them is also a whole number.
Show that among any \(51\) whole numbers, all at most \(100\), there must be two that share no prime factors. For example, \(7\) and \(8\) share no prime factors, and the same is true for \(11\) and \(12\).
There are six kids in the math circle. Each kid has their own seat, and they always sit in the same one. One day, however, the head tutor decided to rearrange the seating, and it turned out that every kid ended up in a different seat from their usual one. In how many ways can the head tutor do this?
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 \(9\).
You are given fifty-one whole numbers. Assume that the square of one number equals the sum of the squares of all the other fifty numbers. Prove that among these fifty-one numbers, there must be an even one.
Show that if \(k\) is a positive whole number, then the decimal expansion of \(1/k\) either has a finite number of decimal places or eventually repeats. For example, \[\frac{1}{5} = 0.2 \qquad\text{or}\qquad \frac{1}{17} = 0.\underbrace{0588235294117647}_{} \underbrace{0588235294117647}_{}\ldots\]
Can you find \(11\) distinct whole numbers whose last digits are all different from each other?