SA Mathematics Olympiad Problem 1998 R2 Q18

The following problem was used as Question 18 in Round 2 of the South African Mathematics Olympiad in 1998. Think about the problem and try to solve it before moving the dynamic sketch below.

SAMO 1998 R2 Q18

(A) A ......... (B) B ......... (C) C ......... (D) D ......... (E) impossible to determine

Challenge
1) Can you solve the problem?
2) Can you explain why (prove that) your solution is correct?

(Note: The above problem can also be situated and contextualized in several different (quasi) real world situations. For example, where should a trainstation be built if the train stops all lie in a straight line?)

SA Mathematics Olympiad Problem 1998 R2 Q18

Formulation
To understand the problem better, we can model the situation with a dynamic geometry sketch as shown above where the position of the houses are indicated by P, Q, R, S, T, and U.
3) Essentially the problem boils down to minimizing the sum of all the distances from a point (a tree) to the houses.

Investigate
4) Click on 'Show Sum Distances' button. Then drag point 'X' and carefully observe the Sum of the Distances as you drag 'X'.
5) What do you notice? Is your earlier solution above correct?

Generalize
6) Can you generalize to any even or odd number of houses?

Proof Challenge
7) Can you prove that your solutions to your generalizations in 6) are correct?

Variation
Suppose there were only 3 houses, but they no longer lie in a straight line, but form a triangle.
8) Where should the children now meet to minimize the sum of the distances they have to walk?
Hint: Go to this webpage for an analogous problem of building an airport for 3 cities which minimizes the sum of the distances of the roads (assuming they are straight) from the cities to the airport: Airport Problem.

SA Mathematics Olympiad 1998 R2 Paper & Solutions
SAMO 1998 R2 Questions & Worked Solutions (PDF)

Reference
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.

External Links
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 27 Feb 2026; updated 2 March 2026.