Subcontests
(5)numbers on 3x556 board
In a board with 3 rows and 555 columns, 3 squares are colored red, one in each of the 3 rows.
If the numbers from 1 to 1665 are written in the boxes, in row order, from left to right (in the first row from 1 to 555, in the second from 556 to 1110 and in the third from 1111 to 1665) there are 3 numbers that are written in red squares.
If they are written in the boxes, ordered by columns, from top to bottom, the numbers from 1 to 1665 (in the first column from 1 to 3, in the second from 4 to 6, in the third from 7 to 9,... ., and in the last one from 1663 to 1665) there are 3 numbers that are written in red boxes.
We call red numbers those that in one of the two distributions are written in red boxes.
Indicate which are the 3 squares that must be colored red so that there are only 3 red numbers.
Show all the possibilities. 3 boxes: blue, white, red, and 9 numberd balls
There are three boxes, one blue, one white and one red, and 8 balls. Each of the balls has a number from 1 to 8 written on it, without repetitions. The 8 balls are distributed in the boxes, so that there are at least two balls in each box. Then, in each box, add up all the numbers written on the balls it contains. The three outcomes are called the blue sum, the white sum, and the red sum, depending on the color of the corresponding box. Find all possible distributions of the balls such that the red sum equals twice the blue sum, and the red sum minus the white sum equals the white sum minus the blue sum.