2008 AIME I Problems/Problem 3
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.
Let the biking rate be , swimming rate be , jogging rate be , all in km/h.
We have . Subtracting the second from twice the first gives . Mod 4, we need . Thus, .
and give non-integral , but gives . Thus, our answer is .
Let , , and be the biking, jogging, and swimming rates of the two people. Hence, and . Subtracting gives us that . Adding three times this to the first equation gives that . Adding four times the previous equation to the first given one gives us that . This gives us that , and then and . Therefore, .
Creating two systems, we get , and . Subtracting the two expressions we get . Note that is odd, so one of is odd. We see from our second expression that must be odd, because is also odd and and are odd. Thus, with this information, we can test cases quickly:
When readdressing the first equation, we see that if will be a multiple of , , we get that and , which works because of integer values. Therefore,
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