Junior Mathematical Olympiad 2025

Instructions

1. Do not open the paper until the invigilator tells you to do so.
2. Time allowed: 2 hours.
3. The use of rulers, set squares and compasses is allowed, but calculators and protractors are forbidden. You are strongly encouraged to use geometrical instruments to construct large, accurate diagrams for geometry problems.
4. Start each question on an official answer sheet on which there is a QR code.
5. If you use additional sheets of (plain or lined) paper for a question, please write the following in the top left-hand corner of each sheet. (i) The question number. (ii) The page number for that question. (iii) The digits following the ':' from the question's answer sheet QR code. Please do not write your name or initials on additional sheets.
6. Write on one side of the paper only. Make sure your writing and diagrams are clear and not too faint. (Your work will be scanned for marking.)
7. Arrange your answer sheets in question order before they are collected. If you are not submitting work for a particular problem, please remove the associated answer sheet.
8. Your answers should be fully simplified, and exact. They may contain symbols such as $\pi$, fractions, or square roots, if appropriate, but not decimal approximations.
9. For Question 1 only, some parts of the question are marked as answer-only. For these parts only, marks will be awarded for the numerical answer with no need for an explanation.
10. For all other questions, and other parts of Question 1, you should give full written solutions, including mathematical reasons as to why your method is correct and complete, including why no other solutions are possible. Just stating an answer, even a correct one, will earn you very few marks; also, incomplete or poorly presented solutions will not receive full marks.
11. Each question is worth up to 10 points. Answers can be awarded partial marks depending on the clarity of the participant's mathematical presentation.
12. To accommodate candidates sitting in other time zones, please do not discuss the paper on the internet until 8am GMT on Thursday 12th June, when the solutions video will be released at ukmt.org.uk/competitions-papers. Candidates in time zones more than 5 hours ahead of GMT must sit the paper on Wednesday 11th June (as defined locally).

Concentrate firstly on one or two questions and then write out full solutions (not just brief 'answers'), including mathematical reasons as to why your method is correct.

You will have done well if you hand in full solutions to two or more questions.

Do not hand in rough work.