It's the other way round : a question which is hard to answer, or a situation which is difficult to resolve, is a problem, This word is usually applied to 'real life', rather than to examination questions.

Science teachers like calling their questions "problems", to be solved, often by a single solution. It is a convenient way to teach. This causes a problem in real life! In real life, questions are not necessarily problems. Questions are just topics for research or discussion, for interest, for attention, for discovery, for creative thinking, etc. Questions do not necessarily require solutions, and often there are multiple or even no answers. Semantic habits affect our view of life. In countries which are run by engineers and where students are divided into mutually exclusive streams of either "science" or "arts", you see the effects of such semantics in politics, policy-making and the social and professional behaviour of young graduates. In Britain, by contrast, for example, many political leaders are graduates in politics and philosophy, or English. You may be interested in comparing the CV of different political leaders, as well as the arts-science mix in the syllabuses of different high school systems in the world.

Exactly, a problem is usually a situation that need to be solved, where a question is something that we have to find the answer.