MathDB

8

Part of 2024 239 Open Mathematical Olympiad

Problems(2)

Combo geo with circles

Source: 239 MO 2024 S8

5/22/2024
There are 2n2n points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into nn pairs and cover each pair of points with a circle containing no other points.
combinatoricsgeometry
Weird binary sequence

Source: 239 MO 2024 J8

5/22/2024
Let x1,x2,…x_1, x_2, \ldots be a sequence of 0,10,1, such that it satisfies the following three conditions: 1) x2=x100=1x_2=x_{100}=1, xi=0x_i=0 for 1≤i≤1001 \leq i \leq 100 and i≠2,100i \neq 2,100; 2) x2n−1=xn−50+1,x2n=xn−50x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50} for 51≤n≤10051 \leq n \leq 100; 3) x2n=xn−50,x2n−1=xn−50+xn−100x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100} for n>100n>100. Show that the sequence is periodic.
combinatorics