MathDB

3

Part of 2024 Serbia Team Selection Test

Problems(2)

Inequality on heptagons (IZhO 2020/3 generalized)

Source: Serbia IMO TST 2024, P3

5/18/2024
Let SS be the set of all convex cyclic heptagons in the plane. Define a function f:S→R+f:S \rightarrow \mathbb{R}^+, such that for any convex cyclic heptagon ABCDEFG,ABCDEFG, f(ABCDEFG)=AC⋅BD⋅CE⋅DF⋅EG⋅FA⋅GBAB⋅BC⋅CD⋅DE⋅EF⋅FG⋅GA.f(ABCDEFG)=\frac{AC \cdot BD \cdot CE \cdot DF \cdot EG \cdot FA \cdot GB} {AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FG \cdot GA}.
a) Show that for any M∈SM \in S, f(M)≥f(∏)f(M) \geq f(\prod), where ∏\prod is a regular heptagon.
b) If f(M)=f(∏)f(M)=f(\prod), is it true that MM is a regular heptagon?
algebra
Config geo with symmedian

Source: Serbia Additional IMO TST 2024, P3 (out of 4)

5/30/2024
Let ABCABC be a triangle with circumcenter OO, angle bisector ADAD with D∈BCD \in BC and altitude AEAE with E∈BCE \in BC. The lines AOAO and BCBC meet at II. The circumcircle of △ADE\triangle ADE meets AB,ACAB, AC at F,GF, G and FGFG meets BCBC at HH. The circumcircles of triangles AHIAHI and ABCABC meet at JJ. Show that AJAJ is a symmedian in △ABC\triangle ABC
geometry