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.
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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.
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.
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.
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?
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.
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?
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?
Our goal at this course is to enhance our students’ mathematical intuition by focusing on a deep understanding of mathematical concepts and to enable them to link different concepts and apply their knowledge to solve mathematical problems to help them to improve their performance at Maths exams.
This course guides you through the fundamentals of Python programming using an interactive Python library known as Turtle.
This course encompasses a range of Geometry topics such as coordinate and spatial geometry, introductory trigonometry, angles, parallel lines, congruent and similar triangles, polygons, circles, the Pythagorean Theorem, and more. Emphasis will be placed on reinforcing Algebra skills and enhancing critical thinking through problem-solving in both mathematical and real-world contexts.
Ask about our courses and offerings, and we will help you choose what works best for you.