MathDB

3

Part of 2012 Czech-Polish-Slovak Match

Problems(2)

ABCD is cyclic implies CFIJ is cyclic

Source: Czech-Polish-Slovak 2012, P3

4/12/2013
Let ABCDABCD be a cyclic quadrilateral with circumcircle ω\omega. Let I,JI, J and KK be the incentres of the triangles ABC,ACDABC, ACD and ABDABD respectively. Let EE be the midpoint of the arc DBDB of circle ω\omega containing the point AA. The line EKEK intersects again the circle ω\omega at point FF (F≠E)(F \neq E). Prove that the points C,F,I,JC, F, I, J lie on a circle.
geometrycircumcirclecyclic quadrilateral
abcd=4, a^2+b^2+c^2+d^2=10, maximize (a+c)(b+d)

Source: 2012 Czech and Slovak, P6

4/12/2013
Let a,b,c,da,b,c,d be positive real numbers such that abcd=4abcd=4 and a2+b2+c2+d2=10.a^2+b^2+c^2+d^2=10. Find the maximum possible value of ab+bc+cd+daab+bc+cd+da.
inequalitiesinequalities proposed