Pentagon

**Sampras** · June 22nd 2009, 08:50 AM

Suppose to each vertex of a pentagon, we assign a number $x_i$ with $s = \sum x_i >0$ . If $x,y,z$ are numbers assigned to three successive vertices and if $y<0$ , then replace $(x,y,z)$ with $(x+y, -y, y+z)$ . Repeat this as long as $y<0$ . Does this algorithm always stop? If it does stop, how many steps are needed? If not, why doesn't it stop? Prove that it stops or does not stop.

**Sampras** · June 25th 2009, 12:08 PM

Can we post solutions if no one has answered it?

**Bruno J.** · June 29th 2009, 08:01 PM

Yes please post a solution to this nice problem! I thought about it a little bit but I couldn't figure it out.

**Sampras** · June 29th 2009, 08:09 PM

It always stops. You need to find a function of the five vertices that decreases when you perform the operation. One such function is $f(x_1, x_2, x_3, x_4, x_5) = \sum_{i=1}^{5} (x_{i}-x_{i+2})^{2}, \ x_6 = x_1, \ x_7 = x_2$ . Suppose $x_4 < 0$ . Then we have $f_{new}-f_{old} = 2sx_4 < 0$ .

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