MathDB

3

Part of 2021 European Mathematical Cup

Problems(2)

10th EMC - Factorials and perfect squares

Source: 10th European Mathematical Cup - Problem J3

12/22/2021
Let ℓ\ell be a positive integer. We say that a positive integer kk is nice if k!+ℓk!+\ell is a square of an integer. Prove that for every positive integer n⩾ℓn \geqslant \ell, the set {1,2,…,n2}\{1, 2, \ldots,n^2\} contains at most n2−n+ℓn^2-n +\ell nice integers. \\ \\ (Théo Lenoir)
factorialPerfect Squarenumber theoryemcEuropean Mathematical Cup
f is N to N, x^2-y^2+2y(f(x)+f(y)) is a perfect square

Source: 10th European Mathematical Cup - Problem S3

12/22/2021
Let N\mathbb{N} denote the set of all positive integers. Find all functions f:N→Nf:\mathbb{N}\to\mathbb{N} such that x2−y2+2y(f(x)+f(y))x^2-y^2+2y(f(x)+f(y)) is a square of an integer for all positive integers xx and yy.
functional equationnumber theoryEuropean Mathematical Cupemc