UKMT

Cayley Mathematical Olympiad

As follow-on Rounds to the Intermediate Maths Challenge, the Cayley, Hamilton and Maclaurin Maths Olympiads are 2 hour Challenges consisting of six Olympiad style problems. Entry to the Intermediate Olympiads is by invitation based on a qualifying IMC score, or by discretionary entry. Around 1,800 students qualify from the IMC each year.

Thank you! Your submission has been received!
Oops! Something went wrong while submitting the form.

Join our newsletter for updates and the Problem of The Day

Boundaries
Instructions

1. Do not open the paper until the invigilator tells you to do so.

2. Time allowed: 2 hours.

3. The use of blank or lined paper for rough working, rulers and compasses is allowed; squared paper, calculators and protractors are forbidden.

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. Do not hand in rough work.

6. Your answers should be fully simplified, and exact. They may contain symbols such as π, fractions, or square roots, if appropriate, but not decimal approximations.

7. You should give full written solutions, including mathematical reasons as to why your method is correct. 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.

Problems

A four-digit number, n, is written as ‘ABCD’ where A, B, C and D are all different odd digits. It is divisible by each of A, B,C and D.

Find all the possible numbers for n.

2023

The diagram shows a triangle ABC with side BA extended to a point E. The bisector of ∠ABC meets the bisector of angle ∠EAC at D.

Let ∠BCA= p and ∠BDA = q.

Prove that p = 2q.

2023

Aroon’s PIN has four digits. When the first digit (readingfrom the left) is moved to the end of the PIN, the resulting integeris 6 less than 3 times Aroon’s PIN. What could Aroon’s PIN be?

2023

The diagram shows a rectangle inside an isosceles triangle. The base of the triangle is n times the base of the rectangle, for some integer n greater than 1.  

Prove that the rectangle occupies a fraction 2/n - 2/n2 of the total area.

2023

The whole numbers from 1 to 2k are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct primefactors.

What is the largest possible value of k?

2023

A bag contains 7 red discs, 8 blue discs and 9 yellow discs. Two discs are drawn at random from the bag. If the discs are the same colour then they are put back into the bag. However, if the discs are different colours then they are removed from the bag and a disc of the third colour is placed in the bag. This procedure is repeated until there is only one disc left in the bag or the only remaining discs in the bag have the same colour.

What colour is the last disc (or discs) left in the bag?

2023

Get ready for olympiads with free problems, extracurricular topics and our courses

Mathematics

Programming

Our courses

Where do you hold your classes?
We hold our classes online or on-site on Saturdays at our branch in Pimlico Academy, London.
You can find our timetable here.
What do you need to start learning online?
For lessons you only need a computer or phone with a microphone, camera and Internet access. Wherever you are - in London, Nottingham, New York or Bali - online lessons will be at hand.
When can I take the introductory lesson?
You can get acquainted with the school at any time convenient for you. To do this, just leave a request and sign up for a lesson.
I can't attend class, what should I do?
It is OK, it happens! Students have the opportunity to cancel a lesson up to 8 hours before the scheduled time without loss of payment. So you can reschedule it for a convenient time, and the teacher will have the opportunity to
I don't have much free time, will I have time to study?
Learning can take place at your own pace. We will select a convenient schedule and at any time we will help you change the schedule, take a break or adjust the program.
How long is one lesson?
All classes last 1 hour.

Contact us

Phone

Mon-Sun from 8am to 9pm
+44 77 07 547 144

Office

Come say hello at our office HQ.
84 Eccleston Square, Pimlico, London SW1V 1NP, UK

...or ask us anything

Smaller 3D model of a student sitting at his desk and preparing for competitions, exams and olympiads.

Ask about our courses and offerings, and we will help you choose what works best for you.

Thank you! Your submission has been received!
Oops! Something went wrong while submitting the form.