MathDB

Problem 2

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

3 symmedians are concurrent

Source: 2023 Viet Nam math olympiad for high school students D1 P2

3/25/2023
Prove that: 33 symmedians of a triangle are concurrent at a point; the concurrent point is called the Lemoine point of the given triangle.
geometry
original polynomial

Source: 2023 Viet Nam math olympiad for high school students D2 P2

3/25/2023
a) Given a prime number pp and 22 polynomialsP(x)=anxn+...+a1x+a0;Q(x)=bmxm+...+b1x+b0.P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0. We know that the product P(x)Q(x)P(x)Q(x) is a polynomial whose coefficents are all divisible by p.p. Prove that: at least 11 in 22 polynomials P(x),Q(x)P(x),Q(x) has all coefficents are all divisible by p.p.
b) Prove that the product of 22 original polynomials is a original polynomial.
algebra