Difference between revisions of "2008 AIME I Problems/Problem 3"

Latest revision as of 20:30, 4 August 2024

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 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, $225+64+25=\boxed{314}$

Solution 4 (Logic)

Building on top of Solution 3, we can add $j+s-2b=17$ and $2b+3j+4s=74$ (sorry, I used different variables) to get $4j+5s=57$ . Logically speaking, most athletic people swim a lot faster than 1 km/h (0.62 mph), so we test out the next case that works, which is 5 km/h (3.1 mph). This seems much more logical, so we plug it into the equation to get $4j+25=57\implies j=8$ . This seems reasonable, as people usually jog at around 8 to 16 km/h. Plugging these values into $2b+3j+4s=74$ , we get $b=15$ . 15 km/h is a little slow (most people bike at 20 km/h), but is still reasonable. So, we get $5^2+8^2+15^2=\boxed{314}$ .

Note: Solving $4j+5s=57$ yields 3 solutions for $(j,s)$ : $(13,1),(8,5),(3,9)$ , and only $(8,5)$ gives us an integer $b$ value and we don't need the assumption, which is also false, since swimming at $5 km/h$ over $4$ hours is also pretty insane, so none of the swimming speeds make any sense.

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 == Solution 4 (Logic) ==
 Building on top of Solution 3, we can add <math>j+s-2b=17</math> and <math>2b+3j+4s=74</math> (sorry, I used different variables) to get <math>4j+5s=57</math>. Logically speaking, most athletic people swim a lot faster than 1 km/h (0.62 mph), so we test out the next case that works, which is 5 km/h (3.1 mph). This seems much more logical, so we plug it into the equation to get <math>4j+25=57\implies j=8</math>. This seems reasonable, as people usually jog at around 8 to 16 km/h. Plugging these values into <math>2b+3j+4s=74</math>, we get <math>b=15</math>. 15 km/h is a little slow (most people bike at 20 km/h), but is still reasonable. So, we get <math>5^2+8^2+15^2=\boxed{314}</math>.
+Note: Solving <math>4j+5s=57</math> yields 3 solutions for <math>(j,s)</math>: <math>(13,1),(8,5),(3,9)</math>, and only <math>(8,5)</math> gives us an integer <math>b</math> value and we don't need the assumption, which is also false, since swimming at <math>5 km/h</math> over <math>4</math> hours is also pretty insane, so none of the swimming speeds make any sense.
 == See also ==

2008 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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