Identifying the original AMM problem sources for two number theory problems

I have been using JSTOR to identify the original year, volume, issue, and problem number of problems that were cited only as “AMM” (American Mathematical Monthly) in various books. My approach has been to search for distinctive phrases and keywords from the problem statements and try to reconstruct the original wording.

This method has been quite successful: out of dozens of AMM-attributed problems that I have investigated, I have been unable to locate only two. I would greatly appreciate any help in identifying their original sources.

A relevant observation is that these two problems are cited merely as “AMM,” but I suspect that both originated in one of the Monthly’s problem sections (elementary or advanced). Every other AMM-attributed problem I have tracked down from the same books turned out to come from a problem section.

The problems are as follows:

Problem 1. Prove that there exist arbitrarily long sequences of consecutive positive integers, none of which can be expressed as the sum of two perfect squares.

This problem appears in the books "Number Theory: Concepts and Problems" and "Problems from the Book".

Problem 2. Let $x$ and $y$ be positive integers such that $x(y+1)$ and $y(x+1)$ are both perfect squares. Prove that exactly one of $x$ and $y$ is a perfect square.

I have seen this problem on AoPS with the attribution “AMM". I am not sure but I think it was also attributed to "AMM" in one of T. Andreescu’s books whose name I do not remember.

Does anyone recognize either of these problems or know their original AMM references (year, volume, issue, and preferably problem number)?