Neverending Maths Competition 2000+

  • Being inspired by the overhelming success of the first two maths competitions we have had organised recently, we decided to continue and also to change the format of the event. Since now, the competition will be run more or less permanently, with possible breaks during examination periods and between semesters (unless we decide to terminate it completly for some reason).

  • We again want to repeat that, from our point of view, for students the point of taking part in the competition is not to defeat others, but to take up the challenge of finding your own way of solving new problems and getting a better insight into the beautiful world of mathematics.

Terms and conditions

  • Everybody can take part in the competition.

  • At the beginning of each week, a new problem may be put on the Web. Either the conditions of the problem or, when formulae or graphs are needed, a link to (a PDF file with the conditions of) the problem will be addded to THIS PAGE.

  • You can attempt any problem. If you want to submit your work, please put it in a legible form on A4 format paper (avoiding using a pencil and/or red and/or green ink/ballpoint pen as a courtesy to potential readers of your work), together with your name, student ID number (if you have one), your current status (e.g. RMIT undegraduate student, 2nd year etc) and the date when you are submitting the solution. Put your submission in the box located on the ground floor of the Richard Berry Building, near the copier opposite room G65. The box is labelled with a red sign with MC-2000 on it. The deadlines for submitting solutions to the problems will be indicated on THIS PAGE.

  • Each semester we will endeavour to determine the winners (participants scoring the maximum total mark for the problems offered during the semester) in the following categories: 1) undergraduate students, 2) honours students, 3) other students, as well as the absolute winner. The winners will most likely be awarded prizes (books or book vouchers), their names will definitely be announced.

  • The judges' decision will be final and no appeals will be considered. No submissions will be returned to participants, and nobody's mistakes will be showed and/or explained, but

  • Solutions to the problems will be published sooner or later.

  • There may be some typos and/or mistakes in the conditions of the problems. Participants can be given extra points for correcting these mistakes!

  • This competition was organised by Kris Wysocki and Kostya Borovkov. We are sorry if some of the problems will prove to be too easy for you. Thanks for participating!

Competition Problems

  • Problem 1. Show that there exists a map on a torus such that to colour it in such a way that regions sharing a common boundary (other than a single point) do not share the same colour, one would need at least seven colours.

    [Problem suggested by Des Robbie. Deadline for submitting solutions is 23 October 2000.]

  • Problem 2. Click HERE to get a PDF file with the problem's conditions.

    [Deadline for submitting solutions is 30 October 2000.]

  • Problem 3. Click HERE to get a PDF file with the problem's conditions.

    [Deadline for submitting solutions is 30 November 2000 - after exams are over!]

  • Problem 4. Let ABC be an equilateral triangle. A point D is such that the distances |AD|=3, |BD|=4, and |CD|=5. Find that area of the triangle ABC.

    [Deadline for submitting solutions is 4 December 2000.]

  • Problem 5. Four convex figures are given on the plane. Show that if every three of them have a non-empty intersection, then the intersection of all four figures is non-empty.

    [Deadline for submitting solutions is 11 December 2000.]

  • Problem 6. Represent the three-dimensional Euclidean space R3 as a union of pairwise disjoint circles* of positive radii.

    *Here: a plane curve every point of which is equidistant from a fixed point within the curve (not a plane figure bounded by such a curve!).

    [Deadline for submitting solutions is 18 December 2000.]

  • Problem 7. Suppose you are given a finite number of points in a plane, and there is no straight line such that all of the points are on the line. Show that there exists a straight line containing exactly two of the points.

    [Deadline for submitting solutions is Christmas-2000.]

Back to Dr Kostya Borovkov's Homepage