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.