# 2008 AIME I Problems/Problem 3

## 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 $74$ 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.

## Solution

### Solution 1

Let the biking rate be $b$, swimming rate be $s$, jogging rate be $j$, all in km/h.

We have $2b + 3j + 4s = 74,2j + 3s + 4b = 91$. Subtracting the second from twice the first gives $4j + 5s = 57$. Mod 4, we need $s\equiv1\pmod{4}$. Thus, $(j,s) = (13,1),(8,5),(3,9)$. $(13,1)$ and $(3,9)$ give non-integral $b$, but $(8,5)$ gives $b = 15$. Thus, our answer is $15^{2} + 8^{2} + 5^{2} = \boxed{314}$.

### Solution 2

Let $b$, $j$, and $s$ be the biking, jogging, and swimming rates of the two people. Hence, $2b + 3j + 4s = 74$ and $4b + 2j + 3s = 91$. Subtracting gives us that $2b - j - s = 17$. Adding three times this to the first equation gives that $8b + s = 125\implies b\le 15$. Adding four times the previous equation to the first given one gives us that $10b - j = 142\implies b > 14\implies b\ge 15$. This gives us that $b = 15$, and then $j = 8$ and $s = 5$. Therefore, $b^2 + s^2 + j^2 = 225 + 64 + 25 = \boxed{314}$.

### Solution 3

Creating two systems, we get $2x+3y+4z=74$, and $2y+3z+4x=91$. Subtracting the two expressions we get $y+z-2x=-17$. Note that $-17$ is odd, so one of $x,y,z$ is odd. We see from our second expression that $z$ must be odd, because $91$ is also odd and $2y$ and $4x$ are odd. Thus, with this information, we can test cases quickly:

When readdressing the first equation, we see that if $2x+3y$ will be a multiple of $6$, $4z \equiv 2 \pmod{6} = 5$, we get that $x=15$ and $y=8$, which works because of integer values. Therefore, $225+64+25=\boxed{314}$

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