MathDB

3

Part of 1989 Greece National Olympiad

Problems(3)

CD >DE if 2AD> AC+ AE

Source: 1989 Greece MO Grade X p3

9/7/2024
From a point AA not on line ε\varepsilon, we drop the perpendicular ABAB on ε\varepsilon and three other not perpendicular lines ACAC, ADAD,AEAE which lie on the same semiplane defines by ABAB, such that (AD)>12((AC)+(AE))(AD )>\frac{1}{2}((AC)+(AE)). Prove that (CD)>(DE).(CD )>(DE). (Points B,C,D,,EB,C,D,,E lie on line ε\varepsilon ) .
geometrygeometric inequality
a^4+ a^3-10 a^2+9 a+4>0 for a>=0

Source: 1989 Greece MO Grade XI p3

9/7/2024
If a≥0a\ge 0 prove that a4+a3−10a2+9a+4>0a^4+ a^3-10 a^2+9 a+4>0.
algebrapolynomialinequalities
x_{n+2}=1/12 x_{n+1}+1/2 x_{n}+1, limit

Source: 1989 Greece MO Grade XII p3

9/7/2024
Find the limit of the sequence xnx_n defined by recurrence relation xn+2=112xn+1+12xn+1x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1 where n=0,1,2,...n=0,1,2,... for any initial values x2,x1x_2,x_1.
limitSequencesreal analysis