Question

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 $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$ hours, swimming for $3$ hours, and biking for $4$ 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.

Answer

Answer: 314

Concepts:system of equations Diophantine Equation rate

Difficulty rating: 2020

Solution:

Let $b,$ $j,$ and $s$ be the biking, jogging, and swimming rates. The two trips give $2b + 3j + 4s = 74$ and $4b + 2j + 3s = 91.$ Doubling the first equation and subtracting the second yields $4j + 5s = 57,$ whose positive integer solutions are $(j, s) = (13, 1),$ $(8, 5),$ and $(3, 9).$

The corresponding values of $2b = 74 - 3j - 4s$ are $31,$ $30,$ and $29,$ so only $(j, s) = (8, 5)$ gives a whole-number rate, $b = 15.$ The sum of the squares is $15^2 + 8^2 + 5^2$ $= 225 + 64 + 25$ $= 314.$

2008 AIME I Problem 3

Answer: 314

Difficulty rating: 2020

Solution:

Problem 3 in Other Years