MathDB

3

Part of 2001 May Olympiad

Problems(2)

numbers on 3x556 board

Source: VII May Olympiad (Olimpiada de Mayo) 2001 L2 P3

9/19/2022
In a board with 33 rows and 555555 columns, 33 squares are colored red, one in each of the 33 rows. If the numbers from 11 to 16651665 are written in the boxes, in row order, from left to right (in the first row from 11 to 555555, in the second from 556556 to 11101110 and in the third from 11111111 to 16651665) there are 33 numbers that are written in red squares. If they are written in the boxes, ordered by columns, from top to bottom, the numbers from 11 to 16651665 (in the first column from 11 to 33, in the second from 44 to 66, in the third from 77 to 99,... ., and in the last one from 16631663 to 16651665) there are 33 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 33 squares that must be colored red so that there are only 33 red numbers. Show all the possibilities.
combinatorics
3 boxes: blue, white, red, and 9 numberd balls

Source: VII May Olympiad (Olimpiada de Mayo) 2001 L1 P3

9/22/2022
There are three boxes, one blue, one white and one red, and 88 balls. Each of the balls has a number from 11 to 88 written on it, without repetitions. The 88 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.
combinatorics