MathDB

2

Part of 2016 Saint Petersburg Mathematical Olympiad

Problems(3)

y_1\le y_2\le...\le y_n, such that x_k \le y_k \le 2x_k for all k=1,2,..., n

Source: St. Petersburg 2016 10.2

5/1/2019
Given the positive numbers x1,x2,...,xnx_1, x_2,..., x_n, such that xi≤2xjx_i \le 2x_j with 1≤i<j≤n1 \le i < j \le n. Prove that there are positive numbers y1≤y2≤...≤yny_1\le y_2\le...\le y_n, such that xk≤yk≤2xkx_k \le y_k \le 2x_k for all k=1,2,...,nk=1,2,..., n
algebrainequalitiespositive real
rooks on surface of a 50x50x50 cube

Source: Saint Petersburg MO 2016 11.2

5/1/2019
The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a 4×4×44 \times 4 \times 4 cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube 50×50×5050 \times 50 \times 50?
maxcubecombinatorics3-Dimensional Geometrycombinatorial geometrygeometry3D geometry
rooks in a 300 x 300 chessboard, min square wanted

Source: St. Petersburg 2016 9.2

5/1/2019
On a 300×300300 \times 300 board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least kk, it is possible to state that there is at least one rook in each k×kk\times k square ?
combinatoricscombinatorial geometrysquareChessboardChess rookRooks