Optimization problems in Geometry — RMO training

Problem 1:

Within a given triangle ABC having all angles less than 120 degrees, determine the point P, so that PA+PB+PC is minimum.

Problem 2:

Within a given convex quadrilateral ABCD, determine the point P, so that PA+PB+PC+PD is minimum.

Problem 3:

In an acute-angled triangle ABC, determine the points D on AB, E on BC, and F on AC so that the perimeter of the triangle DEF is minimum.

Problem 4:

Three cities are located on the vertices of an equilateral triangle of sides 100 km. What must be the minimum total length of the roads connecting these cities so that one can travel from any city to another?

Problem 5:

Four cities are located on the vertices of a square of sides 100 km. What must be the minimum length of the roads connecting these cities so that one can travel from any city to another?

Problem 6:

Consider a park of quadrilateral shape ABCD. A house is located at P on the edge AB. Three more houses are to be built at Q on the edge AD, at R on the edge CD and at S on the edge CB. Locate the points Q, R and S so that the total length of  the road PQRS directly connecting these four houses, constructed within the park, is minimized.

Have some fun with geometry now !

Nalin Pithwa

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