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.
Another possibility, generate a random quiz, using Mathster’s UKMT Mathematics Challenge Online Quiz. Choose Junior, Intermediate or Senior and one of three difficulty levels; you can also choose the number of questions, a time limit and the order the questions are presented in – random or in order of difficulty.
The UK Maths Challenges provide a wonderful library of multiple choice problems; note the these 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 addition for 2015 of extended solutions.
As well as a full solution which does not use the multiple choice answers further ideas are given for investigation.
Links for students, including advice for young mathematicians are provided on Mathematics for Students.
Further outstanding resources from UKMT include the Team Challenge resources. The UK Maths Challenges are included on the Problems & Activities page.
For more sources of multiple choice questions see this post.