Mathcamp 2009 Qualifying Quiz
mathcamp.org
mathcamp.org
1. Forget the "continuous" and break it into half-hour units.
2. Consider what color is the integer a = 7x + 11y. Imagine coloring an infinite 2d grid where each lattice point (x, y) is labeled with 7x + 11y.
3. Let S(n, k) be the number of size-k subsets of [1..n] that contain k.
4. Ask yourself the same question but restrict yourself to only adding powers of 2.
5. Write out the formula for, if a and b are two sides of a triangle, what's the minimum and maximum that the third side can be.
6. Kind of an ugly one. Try writing out the answers for the first few dozen numbers and looking for the patterns. Look at the binary representations.
7. "product" is just a tricky way of obscuring what's going on here. A simpler sequence that obeys this is element n = the log of the nth Fibonacci number. Then you have to know about generating functions or perhaps just Lucas numbers plus cleverness.
8. "10" is relevant because it's (18 / 2) + 1. Try first solving the easier problem where all the lines are restricted to being either vertical or horizontal.
[edit: another minute's thought... actually, the obvious implication of the above is that every odd number is a win, and therefore losses are only those numbers with odd numbers of 2s in their prime factorizations]
In any case, I don't see why these questions belong on Hacker News. They're the sort of fun puzzles that make for a good high school math camp quiz (a clever high school student should figure most of them out in an afternoon), but none of them contains any deeper insight or relevance, as far as I can tell.