MathDB

5

Part of 2024 239 Open Mathematical Olympiad

Problems(2)

Circumscribed quadrilateral and lengths

Source: 239 MO 2024 S5

5/22/2024
A quadrilateral ABCDABCD has an incircle Γ\Gamma. The points X,YX, Y are chosen so that AX−CX=AB−BCAX-CX=AB-BC, BX−DX=BC−CDBX-DX=BC-CD, CY−AY=AD−DCCY-AY=AD-DC and DY−BY=AB−ADDY-BY=AB-AD. Given that the center of Γ\Gamma lies on XYXY, show that AC,BD,XYAC, BD, XY are concurrent.
geometry
3-variable inequality with a weird condition

Source: 239 MO 2024 J5

5/22/2024
Let a,b,ca, b, c be reals such that a2(c2−2b−1)+b2(a2−2c−1)+c2(b2−2a−1)=0.a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0. Show that 3(a2+b2+c2)+4(a+b+c)+3≥6abc.3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.
algebrainequalities proposed