antony-mathematics: problems

An irreducible fraction

Antony — Tue, 03 Nov 2026 00:00:00 GMT

Prove that the fraction is irreducible for every natural number .

A common divisor of two numbers divides every integer combination of them.

Find integers with not depending on .

Solution. Let . Then divides so and the fraction is already in lowest terms.

This was the very first problem of the very first International Mathematical Olympiad, held in Romania in 1959.

One divides another

Antony — Mon, 02 Nov 2026 00:00:00 GMT

Choose any numbers from . Prove that one of the chosen numbers divides another.

Pigeonhole: you need boxes.

Every positive integer can be written as with odd. How many odd are possible here?

Solution. Write each chosen number as with and odd. Since the number is at most , its odd part is one of the odd numbers . There are chosen numbers and only possible odd parts, so two of them share an odd part: and with . Then divides .

The bound is sharp: the numbers contain no such pair, since the ratio of any two of them lies strictly between and . Its easier sibling says that two of the chosen numbers are coprime: two of them must be consecutive. That’s the version Erdős once put to the twelve-year-old Lajos Pósa over dinner, and he solved it on the spot.

The mutilated chessboard

Antony — Sun, 01 Nov 2026 00:00:00 GMT

Two diagonally opposite corner squares are cut from an chessboard, leaving squares. Can the rest be covered exactly by dominoes, each covering two adjacent squares?

Colour the board as a chessboard. What does a single domino always cover?

Solution. No. Every domino covers one black and one white square, so dominoes cover of each. But opposite corners have the same colour, so the cut board has squares of one colour and of the other.

The colouring is an invariant: something every allowed move preserves. Ralph Gomory showed the converse is also neat. Remove any one black and one white square and a tiling always exists: the board’s squares form a single closed loop, and the two holes cut it into two even-length paths.