Problems
Let’s prove that \(1\) is the smallest positive real number: Assume the contrary and let \(x\) be the smallest positive real number. If \(x>1\) then \(1\) is smaller, thus \(x\) is not the smallest. If \(x<1,\) then \(\frac{x}{2}<x\) so \(x\) can not be the smallest either. Then \(x\) can only be equal to \(1\).
Nick writes the numbers \(1,2,\dots,33\), each exactly once, at the vertices of a polygon with \(33\) sides, in some order.
For each side of the polygon, his little sister Hannah writes down the sum of the two numbers at its ends. In total she writes down \(33\) numbers, one for each side.
It turns out that when read in order around the polygon, these \(33\) sums are \(33\) consecutive whole numbers.
Can you find an arrangement of the numbers written by Nick that makes this happen?
Sometimes proving a statement takes careful step-by-step reasoning. But other times, all you need is a single well-chosen example.
Indeed: many problems don’t ask you to prove that something always works. Instead, they ask whether something is possible at all: can an object exist, or can a situation happen, even if it sounds unlikely? In those cases, finding just one example that works is enough to solve the problem.
In this sheet, we’ll practise building examples and constructions. The goal is not to try lots of random cases and hope one works, but to think smartly and strategically about what an example should look like.
By the end of the session, you should feel more confident spotting when an example is enough, and how to construct one that does exactly what the problem asks for.
In how many ways can you read the word TRAIN from the picture below, starting from T and going either down or right at each step?
There are \(100\) people in a room. Each person knows at least \(67\) others. Show that there is a group of four people in this room that all know each other. We assume that if person \(A\) knows person \(B\) then person \(B\) also knows person \(A\).
The numbers \(a\) and \(b\) are integers and the number \(p \ge 3\) is prime. Suppose that \(a+b\) and \(a^2 +b^2\) are divisible by \(p\). Show that \(a^2 + b^2\) is divisible by \(p^2\).
There are \(33\) cities in the Republic of Farfarawayland. The delegation of senators wants to pick a new capital city. They want this city to be connected by roads to every other city in the Republic. They know for a fact that given any set of \(16\) cities, there will always be some city that is connected by roads to all those selected cities. Show that there exists a suitable candidate for the capital.
John drunk a \(\frac16\) of a full cup of black coffee and then filled the cup back up with milk. Then he drunk a third of what he had in the cup. Then, he refilled it back to full with milk again, and after that, drunk a half of the cup. Finally, he once again refilled the cup with milk and drunk everything he had. What did he drink more of - coffee or milk?
A round necklace contains \(45\) beads of two different colours: red and blue. Show that it is possible to find two beads of the same colour next to each other.
Cut this figure into \(4\) identical shapes. (Note: you have to use the entire shape. Rotations and reflections count as identical shapes)
