MathDB

2

Part of 1988 Greece National Olympiad

Problems(3)

AD=AE if AB=AC, <BAD = 2<CDE 1988 Greece MO Grade XI p2

Source:

9/7/2024
In isosceles triangle ABCABC with AB=ACAB=AC, consider point DD on the base BCBC and point EE on side ACAC such that ∠BAD=2∠CDE \angle BAD = 2 \angle CDE. Prove that AD=AEAD=AE.
geometryisoscelesequal segments
BC^2/ MZ^2 >= 8 R U_a / (MD) (ME)

Source: 1988 Greece MO Grade XI p2

9/7/2024
Let ABCABC be a triangle inscribed in circle C(O,R)C(O,R). Let MM ba apoint on the arc BCBC . Let D,E,ZD,E,Z be the feet of the perpendiculars drawn from MM on lines AB,AC,BCAB,AC,BC respectively. Prove that (BC)2(MZ)2≥8RUa(MD)⋅(ME)\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)} where UaU_a is the altitude drawn on BCBC.
geometrygeometric inequality
vector locus |sum MA_i |<=1987, regular 1987-gon

Source: 1988 Greece MO Grade XII p2

9/6/2024
Given regular 19871987 -gon on plane with vertices A1,A2,...,A1987A_1, A_2,..., A_{1987}. Find locus of points M of the plane sych that ∣MA1→+MA2→+...+MA1987→∣≤1987\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987.
vectorgeometryanalytic geometryLocus