In the Problems of Philosophy (PoP) Bertrand Russell proves that there is certain a priori knowledge such as logical principles and mathematical principles. However, does he think these types of knowledge are just analytic or does he prove that they are synthetic?
Russell, in his logicist programme, tries to prove that mathematics is analytic and not synthetic as argued by Kant.
However, he hides his own programme quite successfully in PoP. Maybe, because Russell thinks it should be a book about the problems of philosphy rather than about their solutions. Or, because he thinks the logicist's solution to the problem of the foundation of mathematics would be hard to grasp for the average reader.
Given that, what does he do in the PoP?
Russell's own answer in the PoP consists mainly in the idea that a-priori-knowledge is possible because it is knowledge of universals. Knowledge of universals is a priori as univerals cannot be experienced with our senses (although we can be acquainted with them, according to Russell).
An example of such an a-priori-true-judgement is "Two and two are four". Now, in order for this judgement to be analytic (in the rationalistic tradition) the concept of four would have to somehow involve the concept of two. If this is not the case, the judgement has to be synthetic.
In Chapter VIII Russell considers this decision a serious philosophical problem and gives Kant credit for bringing it up. That is how far he goes in the PoP.
Whereas Kant did not see how concepts like four and two could analytically be linked and went for an answer which explains why mathematical judgements are synthetic, logicists along Russell argue that mathematical truths are logical tautologies and thus analytic.