MathDB

4

Part of 2021 239 Open Mathematical Olympiad

Problems(2)

concurrency lines _|_ plane ABC at Lemoine point, by equilateral triangles

Source: 239 MO 2021 10-11 p4

5/2/2021
Symedians of an acute-angled non-isosceles triangle ABCABC intersect at a point at point LL, and AA1AA_1, BB1BB_1 and CC1CC_1 are its altitudes. Prove that you can construct equilateral triangles A1B1C′A_1B_1C', B1C1A′B_1C_1A' and C1A1B′C_1A_1B' not lying in the plane ABCABC, so that lines AA′,BB′AA' , BB' and CC′CC' and also perpendicular to the plane ABCABC at point LL intersected at one point.
concurrentconcurrencyLemoine pointgeometryperpendicular3D geometry
238^2 (ab)^{239} <1 if a^{239} = ac- 1 , b^{239} = bc- 1 for different a,b,c>0

Source: 239 MO 2021 8-9 p4

5/2/2021
Different positive a,b,ca, b, c are such that a239=ac−1a^{239} = ac- 1 and b239=bc−1b^{239} = bc- 1.Prove that 2382(ab)239<1238^2 (ab)^{239} <1.
algebrainequalities