Below, we present a problem from the 2/18 Algebra and Number Theory set, along with 3 student submitted solution. You may vote up for the solutions that you think should be featured, and should vote down for those solutions that you think are wrong.
Degree 99 Polynomial is a polynomial of degree 99. For exactly 100 (out of 101) integer values ranging from to , we have . Also, . For what value of , is ?
You may try the problem by clicking on the above link.
All solutions may have LaTeX edits to make the math appear properly. The exposition is presented as is, and has not been edited.
About 60% of those who answered this problem got it correct.
Solution A - This solution is completely wrong, despite the votes for it. The biggest tipoff is that it doesn't use the fact that is a polynomial of degree 99. (Yes, it defines as a polynomial of degree 98, but does nothing with it.) While for several values, that tells us nothing about the behavior of , since is not a polynomial, but a rational function. Recall the question Polynomial powered by 2, where knowing that for some values didn't imply that , as pointed out in the discussion.
Note that I often do not penalize typos as I care more about your thought process. However, if your typos carry through or have massive repercussions, then you may be penalized accordingly.
Solution B - This solution is clear in it's presentation, explaining how the function is created, and how to calculate the value of . In this problem, having each of the main equations take up a line increases readability. This solution is presented by Nathan.
Solution C - I had no idea what was happening here. The degree of had nothing to do with the value of . In fact, it is not clear how to calculate the value of , without figuring out what (as defined in Solution B) is, which requires knowing the value of .
Note that is often interpreted as rather than . Be clear in your presentation, and say instead.
Pop quiz: What is the polynomial ?
Note: If you want to submit a problem, please ensure that you the problem is properly phrased, and that you include a proper complete solution. In this case, because the problem was interesting, I figured out my own proper solution.