Problem:
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers kilometers after biking for hours, jogging for hours, and swimming for hours, while Sue covers kilometers after jogging for hours, swimming for hours, and biking for hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
Solution:
Let Ed and Sue's biking rate, let their jogging rate, and let their swimming rate. Then
Adding times the first equation to the second equation and solving for yields . The only ordered pairs that satisfy this equation in which both and are positive integers are , and . However, subtracting the first equation from the second equation and solving for yields , and only the ordered pair produces an integer value (of ) for . Thus the sum of the squares of Ed's rates is .
The problems on this page are the property of the MAA's American Mathematics Competitions