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## Functional equation dependent on a constant

Understand the problem Find all real numbers  for which there exists a non-constant function  satisfying the following two equations for all i)  andii)  Source of the problem Baltic Way 2016 Topic Functional Equations Difficulty Level Easy Suggested Book Functional...

## Pigeonhole principle exercise

Understand the problem We are given  sets of size  each. The size of the intersection of any two sets is exactly . Prove that all the sets have a common element. Source of the problem Austrian-polish mathematical Olympiad 1978 Topic Combinatorics Difficulty Level Easy...

## An interesting arithmetic function on Number Theory using the Prime Numbers

Understand the problem Let  be the set of positive integers. Determine all functions  such that  is divisible by for all positive integers . Source of the problem APMO 2019 Topic Number Theory Difficulty Level Medium Suggested Book An Excursion in Mathematics Start...

## Lottery tickets

Understand the problem Each lottery ticket has a 9-digit number, which uses only the digits 1, 2, 3. Each ticket is coloured red, blue or green. If two tickets have numbers which differ in all nine places, then the tickets have different colours. Ticket 122222222 is...

## A Beautiful Equation using Positive Integers from Singapore Math Olympiad 2008

Understand the problem Find all pairs of positive integers  so that . Source of the problem Singapore MO 2008 Topic Number Theory Difficulty Level Medium Suggested Book An Excursion in Mathematics Start with hints Hint 0Hint 1Hint 2Hint 3Hint 4 Do you really need a...