Some numbers are prettier than others. Suppose everyone in our class were to pick a number between 0 and 100 (inclusive, with decimals allowed). The player closest to pX, where X is the average guess, wins an all expenses paid trip to Sandals Jamaica and a low-mileage, lightly used 2004 PT Cruiser convertible. If multiple students are equally close to pX, they split the prize (don't ask how, just believe). Prove that for any p < 1, the only symmetric NE is for all players to choose zero.