These problems might not be original with me. |

They may fall short of "research" per se, but they illustrate the style of math I like. |

I have answers for them, but I would be interested to see your comments. |

- You have an unbalanced coin whose probability of heads is a real number
*p*between 0 and 1. If you toss the coin a large number of times, what's the probability that the number of heads is congruent to (say) 5 mod 17? (More generally, replace 5 and 17 with other numbers.) - What's the minimum value of
*f*(*t*) = cos*t*+ cos 2*t*+ cos 3*t*+ ... + cos*nt*, and for what value of*t*does it occur? - A balanced
*m*-sided die is rolled*n*times. What's the probability that at least one side comes up at most once? - What's the limit of ( log(
*x*) / log(*x*+1) )^{x}as*x*approaches infinity? - Prove that every positive integer is the sum of two squarefree positive integers.
- Every person owns either 2 cats, 1 cat, or 0 cats. There are
*n*people, and*n*cats, where*n*> 2. Prove that we can simultaneously have each person carry another person's cat. - You have a matrix whose rows are precisely the 2
^{n}distinct row vectors of length*n*whose entries are ±1. A vandal randomly replaces some entries of your matrix with zeros. Prove that some subset of the rows must sum to the zero vector. - Prove that if Σ
*a*_{n}is a divergent series with nonnegative terms, there exists a divergent series Σ*c*_{n}*a*_{n}where*c*_{n}approaches 0. - Characterize the positive integers
*k*such that the equation*x*(*x*+1)=*y*(*y*+*k*) has solutions where*x*and*y*are positive integers.

- A brief note on quantiles in the coupon collector problem.
- A "minimalist" proof that the primes have density zero.
- A self-contained proof that Pell's equation always has nontrivial solutions.
- An analysis of the minimum value and L1 norm of the Dirichlet kernel.
- A proof of (a special case of) the Polya-Vinogradov inequality on character sums.
- An exposition of the proof of Littlewood's conjecture on exponential sums due to McGehee, Pigno, and Smith.
- A calculation of some explicit values of a function related to Chowla's cosine problem. I would be very interested to know whether evaluating this function for an arbitrary n can be reduced to a finite problem (even a prohibitively large finite problem).