Citing exercises in an article
The answer is essentially given in the comments, so let me summarize:
It is a frequent situation that one has to cite an exercise.
It is legitimate. (Polya-Szego is cited > 1400 times according to Mathscinet)
The best thing is to cite a place where the statement is proved, but if you cannot find such a place, citing an exercise is the second best choice.
You can solve the exercise in your paper, or not solve (depending on the difficulty of the exercise and space limitations and other considerations).
And finally my own recommendation: When you refer to an exercise, solve it yourself, no matter whether you include a solution to your paper or not.
Similar considerations apply to handbooks, like Tables of Integrals, etc. They are essentially made for this purpose, but there are sometimes mistakes, not frequently. (Gradshtein-Ryzhik is cited > 2200 times according to Mathscinet, Abramowitz-Stegun 1740.)
I agree wholeheartedly with Alexandre's answer, but there is one other principle I'd add to his list which I believe is essential.
- Whether you merely cite the problem, include some small hints, provide copious hints, or give a full solution, should roughly correspond to the difficulty of the problem.
Of course, you can only know how difficult an exercise is if you have done it for yourself. Some exercises really are easy to experts in the field. Others are extremely difficult. Some are impossible.
And sometimes problems are just wrong. Indeed, one of my papers is a counter-example to the first two exercises in a well-known text.
Here is one additional data point. The standard reference for symmetric function is Macdonald's "Symmetric functions and Hall polynomials". Most of the content of this book is in the exercises; each section of the book contains many pages of useful results and formulas stated without proof. According to Google Scholar, this book has been cited 7735 times, and it seems likely that many (most?) of these citations are references to exercises in the book.