Question

A 33-gon P1 is drawn in the Cartesian plane. The sum of the x-coordinates of the 33 vertices equals 99. The midpoints of the sides of P1 form a second 33-gon, P2. Finally, the midpoints of the sides of P2 form a third 33-gon, P3. Find the sum of the x-coordinates of the vertices of P3.

Answer 1

Label the x-coordinates of P1's vertices x1,...,x33 (cyclic). Then the x-coordinate of the i-th vertex of P2 is yi = (xi + x_{i+1})/2, and that of P3 is
zi = (yi + y_{i+1})/2 = (x_i + 2x_{i+1} + x_{i+2})/4.

Hence
sum_{i=1}^{33} zi = (1/4) sum_{i=1}^{33} (x_i + 2x_{i+1} + x_{i+2})
= (1/4)(S + 2S + S) = S,
where S = sum_{i=1}^{33} x_i = 99.

So the sum of the x-coordinates of P3 is 99.