Subcontests
(3)Merging Monsters
There are N monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than 2.020 times the other monster's weight, we will call this mergence dangerous. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process.Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level.Proposed by houkai Another geometry in Taiwan TST
Let O and H be the circumcenter and the orthocenter, respectively, of an acute triangle ABC. Points D and E are chosen from sides AB and AC, respectively, such that A, D, O, E are concyclic. Let P be a point on the circumcircle of triangle ABC. The line passing P and parallel to OD intersects AB at point X, while the line passing P and parallel to OE intersects AC at Y. Suppose that the perpendicular bisector of HP does not coincide with XY, but intersect XY at Q, and that points A, Q lies on the different sides of DE. Prove that ∠EQD=∠BAC. Proposed by Shuang-Yen Lee