MathDB

6

Part of 2022 Tuymaada Olympiad

Problems(2)

Tuymaada 2022

Source: Tuymaada 2022 Junior P-6

9/12/2022
The city of Neverreturn has NN bus stops numbered 1,2,⋯ ,N.1, 2, \cdots , N. Each bus route is one-way and has only two stops, the beginning and the end. The route network is such that departing from any stop one cannot return to it using city buses. When the mayor notices a route going from a stop with a greater number to a stop with a lesser number, he orders to exchange the number plates of its beginning and its end. Can the plate changing go on infinitely? (K. Ivanov )
graph theorycombinatorics
Tuymaada 2022

Source: Tuymaada 2022 Senior P-6

9/12/2022
Kostya marked the points A(0,1),B(1,0),C(0,0)A(0, 1), B(1, 0), C(0, 0) in the coordinate plane. On the legs of the triangle ABC he marked the points with coordinates (12,0),(13,0),⋯ ,(1n+1,0)(\frac{1}{2},0), (\frac{1}{3},0), \cdots, (\frac{1}{n+1},0) and (0,12),(0,13),⋯ ,(0,1n+1).(0,\frac{1}{2}), (0,\frac{1}{3}), \cdots, (0,\frac{1}{n+1}). Then Kostya joined each pair of marked points with a segment. Sasha drew a 1×n1 \times n rectangle and joined with a segment each pair of integer points on its border. As a result both the triangle and the rectangle are divided into polygons by the segments drawn. Who has the greater number of polygons: Sasha or Kostya? (M. Alekseyev )
analytic geometrycombinatorics geometryalgebra