This list of problems is, for the moment, unsorted. This is a compilation of problems that I am assembling from my drafts.
Problem (~10 points)
Find the GCD of
and for each integer . (Your answer can depend on .)
Problem (~20 points)
Prove that the number of positive divisors of
is odd if and only if is a perfect square.
Problem (~10 points)
What is the last digit of
?
Problem (~25-30 points)
How many numbers
smaller than are there such that is divisible by ?
Problem (~15 points)
Prove that the sequence
, , , … has no perfect squares in it.
Problem (~15 points)
What is the smallest number of separate pieces of wire you need to build the carcass of a cube?
Problem (~20 points)
Show that a graph with
vertices, each of which has degree at least , is connected.
Problem (~15-20 points)
Prove that for a planar connected graph the inequality
holds.
Problem (~20-25 points)
Show that a graph with
vertices, each of which has degree exactly , is not planar.