MathDB

1

Part of 1985 Greece National Olympiad

Problems(2)

1/ sin^2 b, 1/cos^2 b, integer roots of x^2-ax+a=0 - 1985 Greece MO X p1

Source:

9/8/2024
Find all arcs θ\theta such that 1sin⁡2θ,1cos⁡2θ\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta} are integer numbers and roots of equation x2−ax+a=0.x^2-ax+a=0.
algebratrigonometry
E_A x{OA}+E_B x {OB}+E_C x{OC}=O - Carathéodory

Source: 1985 Greece MO Grade XII p1

9/6/2024
Inside triangle ABCABC consider random point OO. Prove that: EAOA→+EBOB→+ECOC→=O→E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O} where EA,EB,ECE_A,E_B,E_C the areas of triangle BOC,COB,AOBBOC, COB, AOB respectively
geometryvectorareasanalytic geometry