the 13th XMO

Part of XES Mathematics Olympiad

Subcontests

(3)

1

Nice Algebra

n\in\mathbb N_+,n\ge 3,a_1,a_2,\cdots ,a_n\in\mathbb R_+.

b_1,b_2,\cdots ,b_n\in\mathbb R_+

satisfy that for

\forall k\in\{1,2,\cdots ,n\},

\sum_{\substack{i,j\in\{1,2,\cdots ,n\}\backslash \{k\}\\i\neq j}}a_ib_j=0.

b_1=b_2=\cdots =b_n=0.

1

An geometry problem

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic.
graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

1

two variable inequality

Find the maximum value of

\lambda ,

\forall x,y\in\mathbb R_+

2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.