2010 Math Prize for Girls, Problems 16-20

Problem #16 This problem involves standard manipulations with power series. In this case, technical issues about convergence are not important, so you can manipulate the expressions much as though they were polynomials and use the fact that if two power series are equal on an interval, then their coefficients are equal.

If you’re having trouble, perhaps a good first place to look is in Ivan Niven’s Mathematics of Choice: How to Count Without Counting. See the chapter on generating polynomials. You can also watch Girls’ Angle Director Lauren Williams simplify the generating function for the Fibonacci sequence in her Girls’ Angle WIM video.

Problem #17 This problem checks whether you are comfortable with manipulation of implicitly defined functions. The way to get comfortable with them is practice, practice, practice.

Problem #18 Sometimes, you just get lucky! When I saw this problem, I actually knew the answer right away, and that is only because just a few weeks ago, Anna found this formula and reported on it in her column Anna’s Math Journal (see pages 8 and 9). But if I hadn’t known about that, the nested square roots are a tip-off to the trigonometric half-angle formulas.

If you’re having trouble, the relevant topic is trigonometry. A good way to learn the trigonometric identities is to take a list of them, and then see if you can reprove every one of them. It may take some going at first, but each time, it’ll get faster. Later, you can try to derive them in alternative ways or with new technology (such as using matrices or complex numbers).

Problem #19 If you can deduce the area of a triangle given the coordinates of its vertices, this problem becomes a fairly straight forward summation. To get the formula for the area, you can use the cross product or a determinant. If you don’t know how to do that, you can also figure out the area by relating the lattice triangle to various rectangles and right triangles. You might enjoy grabbing a mathematically inclined friend one afternoon and seeing just how many different formulas the two of you can find for the area of a triangle in terms of different measurements.

Problem #20 This problem is typical of problems that involve complex roots of unity. Roots of unity have a lot of structure, so they lend themselves well to contest problem construction. The complex nth roots of unity have the symmetry of a regular polygon, form a nice cyclic group, are closed under complex conjugation, and because they are roots of unity, satisfy the nice algebraic identity $z^n = 1$.

If you’re having trouble, play around with complex roots of unity (and complex numbers in general). Any time you spend thinking about this subject is time well spent because complex numbers are important in huge swaths of mathematics.