MathDB

Problem 8

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

inequality with Lemoine point

Source: 2023 Viet Nam math olympiad for high school students D1 P8

3/25/2023
Given a triangle ABCABC with symmedians AD,BE,CFAD,BE,CF concurrent at Lemoine point L(D,E,FL(D,E,F are on the sides BC,CA,AB,BC,CA,AB, respectively).). Prove that: LA+LB+LC≥2(LD+LE+LF).LA+LB+LC\ge 2(LD+LE+LF).
geometry
(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1 is irreducible

Source: 2023 Viet Nam math olympiad for high school students D2 P8

3/25/2023
Prove that: for all positive integers n≥2,n\ge 2, the polynomial(x2−1)2(x2−1)2...(x2−2023)2+1(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1 is irreducible in Q[x].\mathbb{Q}[x].
algebrapolynomial