MathDB

2

Part of 2010 May Olympiad

Problems(2)

equal segments starting with a rectangle and a circle

Source: May Olympiad (Olimpiada de Mayo) 2010 L2

9/11/2018
Let ABCDABCD be a rectangle and the circle of center DD and radius DADA, which cuts the extension of the side ADAD at point PP. Line PCPC cuts the circle at point QQ and the extension of the side ABAB at point RR. Show that QB=BRQB = BR.
geometryrectangleequal segments
find the pattern

Source: XVΙ May Olympiad (Olimpiada de Mayo) 2010 L1 P2

9/22/2022
In stage 00 the numbers are written: 1,11 , 1. In stage 11 the sum of the numbers is inserted: 1,2,11, 2, 1. In stage 22, between each pair of numbers from the previous stage, the sum of them is inserted: 1,3,2,3,11, 3, 2, 3, 1. One more stage: 1,4,3,5,2,5,3,4,11, 4, 3, 5, 2, 5, 3, 4, 1. How many numbers are there in stage 1010? What is the sum of all the numbers in stage 1010?
algebra