28th Eötvös Competition Problems 1924

1.  The positive integers a, b, c are such that there are triangles with sides aⁿ, bⁿ, cⁿ for all positive integers n. Show that at least two of a, b, c must be equal.

2.  What is the locus of the point (in the plane), the sum of whose distances to a given point and line is fixed?

3.  Given three points in the plane, how does one construct three distinct circles which touch in pairs at the three points? 

Solutions

Problem 1

The positive integers a, b, c are such that there are triangles with sides aⁿ, bⁿ, cⁿ for all positive integers n. Show that at least two of a, b, c must be equal.

Solution

Suppose not. There is no loss of generality in taking a < b < c and b = 1. But now for n sufficiently large cⁿ > 2. Then cⁿ > 2 > aⁿ + bⁿ. Contradiction. 

Problem 2

What is the locus of the point (in the plane), the sum of whose distances to a given point and line is fixed?

Solution

Take the line to be L, the point O and the distance sum k. Consider first points P on the same of L as O. Take a line L' parallel to L on the same side and a distance k from L. P meets the condition iff it is a distance k-d from L and hence d from L'. In other words P must be equidistant from O and L'. Hence it must lie on the parabola with focus O and directrix L'. However, also lies between L and L', so the locus is the part of the parabola between the two lines.

Similarly, if P lies on the other side of L' then it must be on the parabola focus O and directrix L", where L" is also parallel to L and a distance k from it. The complete locus is a sort of irregular oval. 

Problem 3

Given three points in the plane, how does one construct three distinct circles which touch in pairs at the three points?

Solution

Take the circumcircle C of the three points. Take the triangle formed by the tangents to the circle at the three points. Its vertices are the required three centers.

For PB = PC, QA = QC, RA = RB.