The UK Maths Challenges provide a wonderful library of multiple choice problems; note the past papers where you will find questions and full solutions. Check also the Individual Challenges pages where you will find details of the challenge and see sample materials. Check the Junior Challenge for example. Note the extended solutions.
As well as a full solution which does not use the multiple choice answers further ideas are given for investigation.
Maths Challenge questions are an excellent resource at any time – not just for preparation for the competitions, particularly with the increased requirement for problem solving skills at all levels. On the outstanding Diagnostic Questions site you can choose Junior Maths Challenge questions by topic by choosing the Themed Quizzes option. These quizzes consist of sets of four or five questions grouped by topic. (Log in to Diagnostic Questions to use the link.)
Very conveniently it is possible to download zip files of questions by topic.
From UKMT, try their free Skills Builders, on-line problem solving questions for both junior and intermediate level. Note the user guide, which applies to both Junior (age 11-13) and Intermediate (age 13-16); this offers clear step by step instructions with screenshots.
UKMT Maths Challenge – Junior
For more sources of multiple choice questions see this post.