MathDB

Problem 6

Part of 2023 VN Math Olympiad For High School Students

Problems(2)

the Lemoine point is the midpoint of the altitude

Source: 2023 Viet Nam math olympiad for high school students D1 P6

3/25/2023
a) Given a triangle ABCABC with ∠BAC=90∘\angle BAC=90^{\circ} and the altitude AH(HAH(H is on the side BC).BC). Prove that: the Lemoine point of the triangle ABCABC is the midpoint of AH.AH.
b) If a triangle has its Lemoine point is the midpoint of 11 in 33 symmedian segments, does that triangle need to be a right triangle? Explain why.
geometry
irreducible in Q[x]

Source: 2023 Viet Nam math olympiad for high school students D2 P6

3/25/2023
Prove that these polynomials are irreducible in Q[x]:\mathbb{Q}[x]: a) xpp!+xp−1(p−1)!+...+x22+x+1,\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1, with pp is a prime number. b) x2n+1,x^{2^n}+1, with nn is a positive integer.
algebrapolynomialnumber theoryprime numbers