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 2t + cos 3t + ... + 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 2ⁿ 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 Σa_n is a divergent series with nonnegative terms, there exists a divergent series Σc_na_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.

Assorted brief informal notes:

• A brief note on quantiles in the coupon collector problem.


• A "minimalist" proof that the primes have density zero.


• A proof of a subtle combinatorial identity.


• 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).

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