MathDB

Problem 5

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

vector equality, Lemoine point

Source: 2023 Viet Nam math olympiad for high school students D1 P5

3/25/2023
Given a triangle ABCABC with Lemoine point L.L. Let a=BC,b=CA,c=AB.a=BC, b=CA,c=AB.
Prove that: a2LA→+b2LB→+c2LC→=0→.{a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .
geometry
Perron's criterion

Source: 2023 Viet Nam math olympiad for high school students D2 P5

3/25/2023
Given a polynomialP(x)=xn+an−1xn−1+...+a1x+a0∈Z[x]P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x] with degree n≥2n\ge 2 and ao≠0.a_o\ne 0. Prove that if ∣an−1∣>1+∣an−2∣+...+∣a1∣+∣a0∣|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|, then P(x)P(x) is irreducible in Z[x].\mathbb{Z}[x].
algebra