Prove that the composition of a tricolorable knot and another knot (except the unknot, whether tricolorable or not) is tricolorable.
I understand that the composition of two tricolorable knots are tricolorable, but I don't know how to approach this problem.
A tricoloring of a knot diagram requires that the three incident strands at a crossing are either all distinct colors or all the same color (and, depending on your definition, usually at least two or all three colors must be used somewhere in the diagram).
Therefore, given a tricolored knot $K$ and any other knot $K'$, we get a nontrivial tricoloring of $K \# K'$ by assigning $K'$ the color of the strand of $K$ onto which it is summed. See the example below.
In fact, it appears that a connected sum of two knots is tricolorable if and only if one of the knots is tricolorable, according to this write-up of an REU project by Kelly Harlan.