Question 177
a) Andrii and Olesia both received a set of cards on which all the integers from $1$ to $2015$ are written. After that Olesia keeps some of the cards (but not all) from her set, and puts the rest aside. Andrii does the same. There are $2015^2$ points on the coordinate plane whose coordinates are integers from $1$ to $2015$. Olesia paints in blue color the points whose first coordinate equals the number on one of her cards and whose second coordinate equals the number on one of Andrii's cards, and Andrii paints in yellow color the points whose first coordinate equals the number on one of his cards and whose second coordinate equals the number on one of Olesia's cards (some points can be painted in both yellow and blue colors). Prove that no matter which cards Andrii and Olesia choose, they cannot make all the $2015^2$ points painted in at least one color.
b) Oxana joins Andrii and Olesia and takes all the cards that Olesia and Andrii put aside. Then they paint the coordinate plane with the following rule: Olesia paints in blue color the points whose first coordinate equals the number on one of her cards and whose second coordinate is on one of Andrii's cards, Andrii paints in yellow color the points whose first coordinate equals the number on one of his cards and whose second coordinate is on one of Oxana's cards, and Oxana paints in green color the points whose first coordinate is on one of her cards and whose second coordinate is on one of Olesia's cards (again some points can be painted in more than one color). How do Olesia and Andrii have to choose their cards in order to paint every point of the $2015^2$ points in at least one color?
b) Oxana joins Andrii and Olesia and takes all the cards that Olesia and Andrii put aside. Then they paint the coordinate plane with the following rule: Olesia paints in blue color the points whose first coordinate equals the number on one of her cards and whose second coordinate is on one of Andrii's cards, Andrii paints in yellow color the points whose first coordinate equals the number on one of his cards and whose second coordinate is on one of Oxana's cards, and Oxana paints in green color the points whose first coordinate is on one of her cards and whose second coordinate is on one of Olesia's cards (again some points can be painted in more than one color). How do Olesia and Andrii have to choose their cards in order to paint every point of the $2015^2$ points in at least one color?