4
Part of 2023 May Olympiad
Problems(2)
foldina a rectangle paper 3 times
Source: 2023 May Olympiad L2 p4
3/24/2024
Matías has a rectangular sheet of paper , with .Initially, he folds the sheet along a straight line , where is a point on the side , so that vertex is located on side , as shown in the figure. Then folds the sheet again along a straight line , where is a point on side , so that vertex lies on the line ; and finally folds the sheet along the line . Matías observed that the vertices and were located on the same point of segment after making the folds. Calculate the measure of the angle .
https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png
geometryrectanglepaperfolding
numbers in 3x2023 board (2023 May Olympiad L1 p4)
Source:
3/24/2024
There is a board with three rows and columns. In the first row the numbers are written from to , ordered from least to greatest. The devil writes those same numbers in the boxes in the second row, but ordered to his choice. Then, in each box in the third row he writes the difference between the two numbers already written in his own column (the largest minus the smallest). For example, if the first two boxes of a column are the numbers and , in the third box it is written . Explain why, no matter how the devil completed the second row of the board, it will never happen that multiplying them numbers in the third row the result is odd.
combinatorics