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 Σan is a divergent series with nonnegative terms, there exists
a divergent series Σcnan where cn
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 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).