Into which a) two parts, b) n parts should a given number be divided so that the product of the parts is the greatest?
Prove that the sum of a) two, b) several numbers with a fixed product becomes smallest when the numbers are equal.
Let x₁ < x₂ < ... < xₙ. Find such a point x on the number line that the sum of the distances from x to xᵢ is minimised.
(Geometry problem) Find the minimum value of the expression: square root of (y - 2)² + 1 + square root of (x² + y²) + square root of (x - 2)² + 4.
Prove that among a) rectangles, b) rhombuses with the same perimeter, the square has the greatest area. c) What should be the sides of a rectangle inscribed in a circle for its area to be the greatest?
Among all triangles ABC with given sides AB and BC, find the triangle with the largest area.
A straight line l and two points A and B are given, lying a) on opposite sides, b) on the same side of the line. Find such a point X on line l that AX + BX is minimised.
A ray of light travels from point A to point B, reflecting off a flat mirror a. Prove that, according to the law of reflection (angle of incidence equals angle of reflection), the ray chooses the shortest path.
A polygon is given, symmetric with respect to point O. Prove that for this point, the sum of distances to the vertices of the polygon is minimised.
(Fermat–Torricelli problem) Find the point for which the sum of distances to the vertices of a given triangle is minimised.
(Fangyuan’s problem) A triangle ABC is given, which is acute-angled. For which points K, L, and M, lying on sides BC, AC, and AB respectively, is the perimeter of triangle KLM minimised?
(Dido’s problem) Prove that among all shapes with a given perimeter, the one with the largest area is the circle.
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 trial 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.
What should I expect from the trial lesson?
The trial lesson is a 30-minute online session designed to get a sense of how your child approaches mathematical thinking and problem solving. (In practice, it often runs a bit longer if the student is engaged!)
We typically explore a range of fun and challenging problems drawn from competitions. We adapt the difficulty based on how the student responds, aiming to make it both accessible and stimulating.
After the session, we’ll have a quick conversation with the parent to share observations and suggest a personalised path forward.
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.
Meet our team
Our teachers will tell you how to prepare for exams, help you cope with difficult tasks and win the Olympiad They will tell you about the pitfalls of exams and the most common mistakes, and explain how to avoid them
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