MathDB

3

Part of 2024 Brazil National Olympiad

Problems(2)

largest k increasing path that is garanteed

Source: Brazilian Mathematical Olympiad 2024, Level 2, Problem 3

10/12/2024
The numbers from 11 to 100100 are placed without repetition in each cell of a 10×1010 \times 10 board. An increasing path of length kk on this board is a sequence of cells c1,c2,…,ckc_1, c_2, \ldots, c_k such that, for each i=2,3,…,ki = 2, 3, \ldots, k, the following properties are satisfied:
• The cells cic_i and ci−1c_{i-1} share a side or a vertex; • The number in cic_i is greater than the number in ci−1c_{i-1}.
What is the largest positive integer kk for which we can always find an increasing path of length kk, regardless of how the numbers from 1 to 100 are arranged on the board?
combinatoricsboard
minimum number of bisector in a convex polygon

Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 3

10/12/2024
Let n≥3 n \geq 3 be a positive integer. In a convex polygon with n n sides, all the internal bisectors of its n n internal angles are drawn. Determine, as a function of n n , the smallest possible number of distinct lines determined by these bisectors.
geometrycombinatorial geometryangle bisectordiagonals