Mock USAMO by probability1.01 dropped problems
Let be a fixed positive integer, and let be distinct positive integers. We define . Prove that there are no distinct positive integers for which is a geometric sequence.
Reason: The result is somewhat interesting, but no clever or surprising steps are used to solve the problem. Solution
In triangle , , let the incircle touch , , and at , , and respectively. Let be a point on on the opposite side of from . If and meet at , and and meet at , prove that , , and concur.
Reason: The whole incircle business seemed rather artificial. Besides, it wasn't that difficult.
In triangle , let be an interior point. Suppose the circumcircles of and intersect again at and respectively. Prove that iff .
Reason: This is really easy with inversion. It's also quite hard without ideas from inversion (try to find a way!). Too bad, it was a pretty nice problem otherwise.
Let be a cyclic quadrilateral. Prove that
Reason: This problem was interesting but too simple.
Let a sequence be defined by and . Prove that all numbers in the sequence are integers.
Reason: This was actually a pretty good problem, but it was vying for the number 1 or number 4 spot with lots of other problems. Plus, bubala made this one.
In the game of Laser Gun, two players move along the -axis, and a mirror lies along the segment connecting and . A number of opaque computer-controlled tiles of width 1 unit can slide back and forth along the mirror. Each player tries to shoot a laser at himself by reﬂecting it off of the mirror, thus scoring a point. The computer moves its opaque tiles to try to block the shots. If the players each move at twice the speed of each tile, then what is the minimum number of tiles needed to ensure that neither player can ever score a point?
Reason: The wording is way too confusing, and the whole shooting yourself thing didn’t do Laser Gun justice. However, if you do actually understand this problem, it's rather interesting.