2008 AIME I Problem 3

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

Small hint:

The two trips give

2b + 3j + 4s = 74

and

4b + 2j + 3s = 91

in whole numbers

Big hint:

Eliminate

b

to get

4j + 5s = 57;

of its three positive solutions, only one makes

b

a whole number

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.