MathDB

4

Part of 2003 May Olympiad

Problems(2)

2003 green points are contained in a triangle T of area <4

Source: IX May Olympiad (Olimpiada de Mayo) 2003 L2 P4

9/19/2022
Bob plotted 20032003 green points on the plane, so all triangles with three green vertices have area less than 11. Prove that the 20032003 green points are contained in a triangle TT of area less than 44.
geometrycombinatoricscombinatorial geometryareas
list of numers game

Source: IX May Olympiad (Olimpiada de Mayo) 2003 L1 P4

9/22/2022
Celia chooses a number nn and writes the list of natural numbers from 11 to nn: 1,2,3,4,...,n−1,n.1, 2, 3, 4, ..., n-1, n. At each step, it changes the list: it copies the first number to the end and deletes the first two. After n−1n-1 steps a single number will be written. For example, for n=6n=6 the five steps are: 1,2,3,4,5,6→3,4,5,6,1→5,6,1,3→1,3,5→5,1→5 1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5 and the number 55 is written. Celia chose a number nn between 10001000 and 30003000 and after n−1n-1 steps the number 11 was written. Determine all the values of nn that Celia could have chosen. Justify why those values work, and the others do not.
combinatoricsnumber theory