MathDB

8

Part of 2015 All-Russian Olympiad

Problems(2)

Find all values of N

Source: All Russian Grade 9 Day 2 P4

12/12/2015
N≥9N\geq9 distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than 11. It happens that for very 88 distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values NN for which this is possible. (F. Nilov)
combinatorics
Coloring Cells of Graphs

Source: All Russian Olympiad 2015 11.8

12/11/2015
Given natural numbers aa and bb, such that a<b<2aa<b<2a. Some cells on a graph are colored such that in every rectangle with dimensions A×BA \times B or B×AB \times A, at least one cell is colored. For which greatest α\alpha can you say that for every natural number NN you can find a square N×NN \times N in which at least α⋅N2\alpha \cdot N^2 cells are colored?
combinatorics