5
Part of 2021 OMpD
Problems(2)
Find the bad apple with the minimum number of moves. Beautiful Problem.
Source: 2021 2nd OMpD L3 P5 - Brazil - Olimphída Matemáticos por Diversão
7/8/2023
Snow White has, in her huge basket, apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous.To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die.(a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days.(b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?
combinatoricsgame strategy
angle wanted, perpendicular on angle bisector, summetric point, circumcircle
Source: 2021 2nd OMpD L2 P5 - Brazil - Olimpíada Matemáticos por Diversão
9/23/2022
Let be a triangle with and with . Let be the internal bisector of and let be the perpendicular, through , on . Denote by the intersection of and , and denote by the intersection of with the segment . Let also be the symmetric of with respect to the line . Assuming that the circumcircle of triangle is tangent to line and lies on , determine the value of .
geometryanglescircumcircle