If one mentions that one is perfecting a new device to a customer and mention that the problem has been solved, would this be considered to be disclosure and trigger the one-year clock to file a provisional patent?
13 U.S.C. 103 describes how public prior art may eliminate the patentability of an invention:
A patent for a claimed invention may not be obtained... if the differences between the claimed invention and the prior art are such that the claimed invention as a whole would have been obvious before the effective filing date of the claimed invention to a person having ordinary skill in the art to which the claimed invention pertains.
The key criterion here is whether the information you publicly disclosed (the prior art) made "the claimed invention as a whole... obvious..." In particular, it must be obvious to an expert in the domain of patent, i.e., "a person having ordinary skill in the art to which the claimed invention pertains."
If you've only disclosed that a solution for Problem X does positively exist, you probably have not given away information that makes the particular details of your solution "obvious" to anyone, including experts in the field of Problem X.