parmenides51 wrote: Given a circle centered at the origin. Prove that there is a circle of smaller radius on which lies at least points with integer coordinates. Sorry, I don't understand, but what does "lies at least points with integer coordinates" mean?

it's probably ''lie'' and not ''lies'' , (I use google translate and check whether it makes sense) I believe that it means that from all the circles that contain at their circumference points with integer coordinates of center (0,0), there is one having the smallest radius this is the original Russian wording: Quote: Дана окружность с центром в начале координат. Докажите, что найдется окружность меньшего радиуса, на которой лежит не меньше точек с целыми координатами source

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.