MathDB

2

Part of 2022 European Mathematical Cup

Problems(2)

Floor function is back, yay!

Source: European Mathematical Cup 2022, Junior Division, Problem 2

12/19/2022
Find all pairs (x,y)(x,y) of positive real numbers such that xyxy is an integer and x+y=⌊x2−y2⌋x+y = \lfloor x^2 - y^2 \rfloor.
algebrafloor functionfunction
As some nations like to say "Heavy theorems mostly do not help"

Source: European Mathematical Cup 2022, Senior Division, Problem 2

12/20/2022
We say that a positive integer nn is lovely if there exist a positive integer kk and (not necessarily distinct) positive integers d1d_1, d2d_2, …\ldots, dkd_k such that n=d1d2⋯dkn = d_1d_2\cdots d_k and di2∣n+did_i^2 \mid n + d_i for i=1,2,…,ki=1,2,\ldots,k.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than 11, which is a perfect square of an integer?
number theoryDivisorsgreatest common divisorDivisibility