I was reviewing my Anki flashcards from my second year discrete maths course (MATH2302) and rediscovered this fact for myself: "For any integer, the number of odd partitions is the same as the number of distinct partitions." I found this strange; it was clear from the generating functions, but I didn't have any intuition on why this would be the case.
To give some background, a partition is a way to write an integer as a sum of positive integers where we ignore ordering ...
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Recently MSS held its integration bee (think spelling bee, but integration questions) for 2023 and I was set aside the task of creating the questions. I always like this job (I did it last year as well), as coming up with good integrals is tricky. The aim is that the integrals can be done quickly if you know what you are doing (say in less than one minute), achieveable with only the basic integration techniques (substitution and integration by parts, and avoiding trig substitutions and hyperbolic functions if possible), but still can stump some contestants and wow the audience. ...
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