plaintainmountain

Please do not answer this question. It is from AoPS Beast Academy.

Alternatively you could solve for a polynomial whose roots are \(p+3\) and \(q+3\) and then use Vieta's. 

This answer is ChatGPT-generated and incorrect. I believe you have already been told to stop using AI's to answer questions. If you can't figure it out, that's fine but don't spread misinformation using ChatGPT. SpectraSynth is currently going to town spamming AI answers. Don't contribute to the problem.

For any pair of adjacent cards, the probability that they are both hearts is \(\frac{13}{52} \cdot \frac{12}{51}=\frac1{17}\). Now use linearity of expectation.

15 and 51, 24 and 42, and 33 and 33. Of these solutions, one makes sense, One is kinda sketchy, and the last one is absurd.

It will take you infinite moves since you will never actually "reach" the door. 

The angle will still be \(38^{\circ}\) regardless of magnification. 

Here is a hint: inscribe the octagon in a rectangle and subtract the excess area