MathDB

Problem 10

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

angle condition with Lemoine point

Source: 2023 Viet Nam math olympiad for high school students D1 P10

3/25/2023
Given a triangle ABCABC with Lemoine point L.L. Choose points X,Y,ZX,Y,Z on the segments LA,LB,LC,LA,LB,LC, respectively such that:∠XBA=∠YAB,∠XCA=∠ZAC.\angle XBA=\angle YAB,\angle XCA=\angle ZAC. Prove that: ∠ZBC=∠YCB.\angle ZBC=\angle YCB.
geometry
(x(x+1)(x+2)(x+3))^{2^{2023}}+1is irreducible in Q[x]

Source: 2023 Viet Nam math olympiad for high school students D2 P10

3/25/2023
Prove that: the polynomial(x(x+1)(x+2)(x+3))22023+1(x(x+1)(x+2)(x+3))^{2^{2023}}+1is irreducible in Q[x].\mathbb{Q}[x].
algebra