MathDB

2

Part of 1983 Bundeswettbewerb Mathematik

Problems(2)

right triangle construction given R and r

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

11/22/2022
The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.
geometryinradiuscircumradiusright triangleconstruction
2 player game, removing 512, 256, 128, ..., 1 numbers from {0,1,..., 1024}

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

11/22/2022
Two people AA and BB play the following game: They take from {0,1,2,3,...,1024}\{0, 1, 2, 3,..., 1024\} alternately 512512, 256256, 128128, 6464, 3232, 1616, 88, 44, 22, 11, numbers away where AA first removes 512512 numbers, BB removes 256256 numbers etc. Two numbers a,ba, b remain (a<ba < b). BB pays AA the amount bāˆ’ab - a. AA would like to win as much as possible, BB would like to lose as little as possible. What profit does AA make if does every player play optimally according to their goals? The result must be justified.
combinatoricsnumber theory