MathDB

8

Part of 2017 Tuymaada Olympiad

Problems(2)

4 cops and a robber

Source: Tuymaada 2017 Junior Level

7/18/2017
We consider the graph with vertices A1,A2,…A2015A_1,A_2,\dots A_{2015} , B1,B2,…B2015B_1,B_2,\dots B_{2015} and edges AiAi+1,AiBi,BiBi+17A_iA _{i+1}, A_iB_i, B_iB_{i+17} , taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.)
Tuymaada 2017 Q8 Juniors
combinatorics
locus with coordinates from distances about 2 points

Source: Tuymaada 2017 Seniors Q8 (day 2 P4)

9/26/2017
Two points AA and BB are given in the plane. A point XX is called their preposterous midpoint if there is a Cartesian coordinate system in the plane such that the coordinates of AA and BB in this system are non-negative, the abscissa of XX is the geometric mean of the abscissae of AA and BB, and the ordinate of XX is the geometric mean of the ordinates of AA and BB. Find the locus of all the preposterous midpoints of AA and BB.
(K. Tyschu)
Locuscoordinate geometryanalytic geometrygeometry