2007 SMT General Round Problem 2

Problem

Aliens from Lumix have one head and four legs, while those from Obscra have two heads and only one leg. If 60 aliens attend a joint Lumix and Obscra interworld conference, and there are 129 legs present, how many heads are there?

Solution

Let's say there are $x$ aliens from Lumix and $y$ aliens from Obscra. Because there are 60 aliens that attended the meeting, we have $x+y=60$ . Because every alien from Lumix has $4$ legs, every alien from Obscra has $1$ leg, and there are $129$ legs present, we must have $4x+1y=129$ . After subtracting equation one from equation two, we get $3x=69$ , so $x=23$ . Plugging that into equation one gives $y=37$ , so we have $23$ aliens from Lumix and $37$ aliens from Obscra. Because every alien from Lumix has $1$ head and every alien from Obscra has $2$ heads, the total number of neads would be $1\times23+2\times37=23+74=97$ , so our answer is $\boxed{\mathrm{97}}$ .

~Yuhao2012

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2007 SMT General Round Problem 2

Problem

Solution