Forty Minutes No Ideas

You are 40 minutes into a problem. In the last 10 of those you produced two new ideas and rejected both, and you have nothing else queued. The USACO Guide's contributors give different numbers: Darren Yao says read the solution after 15 to 20 minutes of no meaningful progress at the entry tier and 30 to 40 one tier up, Nathan Chen says anywhere from 1 to 5 hours, William Lin says roughly as long as you would have in a real contest.

What does their disagreement actually resolve to?

Read the editorial, understand the key idea, and move on to the next problem. Implementing a solution you have already understood is typing practice.

The clock is a proxy; the signal is whether you are still generating ideas. Benjamin Qi's rule is to keep thinking while new ideas are still arriving and stop when they are not. You are past that point, so read the editorial, and then implement it yourself rather than only understanding it.

Yao's numbers are the calibrated ones because they are tier-specific: hold to 30 to 40 minutes and stop the moment the timer rings, regardless of what you were thinking about.

Never read the editorial. A problem you looked up is a problem you did not learn, and the difference between competitors is tolerance for being stuck.