+24 1. A regular dodecagon \(P_1 P_2 P_3 \dotsb P_{12}\) is inscribed in a circle with radius 1. Compute
\((P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\)(The sum includes all terms of the form \((P_i P_j)^2\), where \(1 \le i < j \le 12.\))
2. In the diagram, \(\angle U = 30^\circ\), arc XY is \(170^\circ\), and arc VW is \(110^\circ\). Find arc WY, in degrees. (view diagram)![[asy] unitsize(2 cm); pair A, B, C, D, E, F; D = dir(160); B = dir(200); E = dir(0); C = dir(270); A = extension(B,C,D,E); F = extension(B,E,C,D); draw(Circle((0,0),1)); draw(C--A--E); draw(C--D--B--E); label(](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2Fa%2Fc%2Fa%2Faca12b1fb2644f794294af3b15cfeab348364d6f.png)
Thank you!
1. We can use the sine law to find P1 P2
By the sine law, P1 P2 = sin 15. We can also find P1 P_3 = sin 60, P1 P4 = sin 90, etc.
So (P1 P2)^2 + (P1 P3)^2 + … + (P11 P12)^2 = 12 (sin 15)^2 + 12 (sin 30)^2 + 12 (sin 45)^2 + … + 12 (sin 90)^2 = 42.
2. Arc WY - Arc XV = Angle U = 30 degrees.
Arc WY + Arx XV = 360 - 170 - 110 = 80 degrees.
Therefore, arc WY = (30 + 80)/2 = 55 degrees.
