MathDB

3

Part of 2023 May Olympiad

Problems(2)

junior area chasing, 3 equal segments, 2 equilaterals (2023 May Olympiad L1 p3)

Source:

3/24/2024
On a straight line ℓ\ell there are four points, AA, BB, CC and DD in that order, such that AB=BC=CDAB=BC=CD. A point EE is chosen outside the straight line so that when drawing the segments EBEB and ECEC, an equilateral triangle EBCEBC is formed . Segments EAEA and EDED are drawn, and a point FF is chosen so that when drawing the segments FAFA and FEFE, an equilateral triangle FAEFAE is formed outside the triangle EADEAD. Finally, the lines EBEB and FAFA are drawn , which intersect at the point GG. If the area of triangle EBDEBD is 1010, calculate the area of triangle EFGEFG.
geometryareas
deleting multiple or divisor in pairs from 2-50 on a blackboard

Source: 2023 May Olympiad L2 p3

3/24/2024
The 4949 numbers 2,3,4,...,49,502,3,4,...,49,50 are written on the blackboard . An allowed operation consists of choosing two different numbers aa and bb of the blackboard such that aa is a multiple of bb and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.
combinatoricsnumber theory