MathDB

2

Part of 2022 Saudi Arabia JBMO TST

Problems(2)

A, B', P, Q concyclic wanted, 2 altitudes and 2 circles related

Source: 2022 Saudi Arabia JBMO TST 1.2

11/1/2022
Let BB′BB', CC′CC' be the altitudes of an acute-angled triangle ABCABC. Two circles passing through AA and C′C' are tangent to BCBC at points PP and QQ. Prove that A,B′,P,QA, B', P, Q are concyclic.
geometryconcurrency
max \sqrt{b^2+c^2}/(3-a)+ \sqrt{c^2+a^2}/(3-b)+a+b-2022c if a^2+b^2+c^2=2

Source: 2022 Saudi Arabia JBMO TST 2.2

11/3/2022
Consider non-negative real numbers a,b,ca, b, c satisfying the condition a2+b2+c2=2a^2 + b^2 + c^2 = 2 . Find the maximum value of the following expression P=b2+c23−a+c2+a23−b+a+b−2022cP=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c
inequalitiesmaxalgebra