Theory
The Extreme Principle is a problem-solving method where you focus on the largest, smallest, or otherwise most “extreme” element in a set to gain insight into the problem. By examining such an element, you can often find a contradiction, reduce the complexity, or uncover a key property that must hold.
For example, if you are trying to prove that a certain configuration is impossible, you might start by assuming it exists and then pick the “extreme” case — the point farthest to the left, the smallest number, the shortest segment — and show that this choice leads to a contradiction. In geometry and combinatorics, the Extreme Principle often helps identify hidden constraints that are hard to see when looking at the problem as a whole.
Problems
- Is it possible to mark 100 points on a plane so that each marked point is the midpoint of the segment connecting two other marked points? What about in space?
- The sum of positive numbers x₁, x₂, …, x₁₀₀ is equal to 1. Prove the inequality x₁x₂ + x₂x₃ + … + x₉₉x₁₀₀ ≤ 1/4.
- Prove that there is no convex polyhedron in which all faces have a different number of sides.
- On the plane, 100 straight lines are drawn. No two are parallel, and no three pass through the same point. Into how many regions do the lines divide the plane?
- a) Several identical coins are lying on a table without overlapping. Prove that there is a coin that touches no more than three others. b) There are 21 numbers, and the sum of any five of them is positive. Prove that the sum of all the numbers is positive.
- Six numbers are arranged around a circle, and each number is equal to the absolute value of the difference of the next two numbers in clockwise order. The sum of all the numbers is equal to 1. Find these numbers.
- Ali-Baba is trying to get into a cave. At the entrance stands a square table with a vessel in each corner. In each vessel there is a herring, which may be placed head-up or tail-up. From the outside, the positions of the herrings are not visible. Ali-Baba may put his hands into any two vessels, feel how the herrings are positioned, and set them however he likes (he may leave them as they were or flip one or both). This operation may be repeated several times. However, after each move the table is spun rapidly, so when it stops, it is impossible to tell which vessels were previously touched. The cave door opens if all herrings are in the same position. Help Ali-Baba find a strategy to enter the cave.
- Tower of Hanoi. The “Tower of Hanoi” puzzle consists of three pegs, with seven rings of decreasing size stacked on one of them. It is allowed to remove one ring at a time from any peg and place it on any other peg, but it is forbidden to place a larger ring on top of a smaller one. Is it possible, following these rules, to transfer all rings to another peg?
- On an infinite squared sheet of paper, 100 cells are coloured black and all others white. In one move, you may switch the colour of any four cells forming a 2×2 square. Prove that it is possible to make all cells white in several moves if and only if every row and every column contains an even number of black cells.
- In each cell of an infinite squared sheet of paper, a natural number is written. It turns out that each number is equal to the arithmetic mean of its four neighbouring numbers. Prove that all the numbers are equal to each other.