Solutions to the problems exist in notebooks made by the professors who taught the Introductory Physics course at Caltech in the 1960s (though of course, those are not the only possible solutions). Copies of these notebooks were entrusted to me and Rudolph Pfeiffer, editors of the book Exercises for The Feynman Lectures on Physics in which most, if not all, the problems that appear in the Original Course Handouts published at The Feynman Lectures Website can be found. However, these notebooks were given only for the purpose of checking the solutions in them, to ensure the answers published in the exercise book are all correct. Moreover, they were given under the strict condition that the solutions would never be published or shared, because the Caltech professors who compiled the notebooks feel strongly that students can only learn something from the problems if they create their own solutions, and that showing students solutions to problems is counterproductive because it robs the problems of their power to teach, rendering them useless. We publish the answers to the problems in the back of the exercise book so that students can check whether their own solutions produce the same (verified) answers.
That being said, I might add that we have in fact published solutions to a minority of the problems in the exercise book - the problems that ask students to demonstrate something is true. In this case the "answer" is given and the only possible response is a solution. All such problems for which a brief (one paragraph) solution is known are solved in the back of the exercise book.