MathDB

Problem 4

Part of 2022 VN Math Olympiad For High School Students

Problems(2)

vector and lengths (in)equality

Source: 2022 Viet Nam math olympiad for high school students D1 P4

3/20/2023
Assume that △ABC\triangle ABC is acute. Let a=BC,b=CA,c=ABa=BC, b=CA, c=AB. a) Denote HH by the orthocenter of △ABC\triangle ABC. Prove that:a.HA→HA+b.HB→HB+c.HC→HC=0→.a.\frac{{\overrightarrow {HA} }}{{HA}} + b.\frac{{\overrightarrow {HB} }}{{HB}} + c.\frac{{\overrightarrow {HC} }}{{HC}} = \overrightarrow 0 . b) Consider a point PP lying on the plane. Prove that the sum:aPa+bPB+cPCaPa+bPB+cPC get its minimum value iff P≡HP\equiv H.
geometryvector
determine k(2),k(4),k(5),k(10)

Source: 2022 Viet Nam math olympiad for high school students D2 P4

3/22/2023
Given Fibonacci sequence (Fn),(F_n), and a positive integer mm, denote k(m)k(m) by the smallest positive integer satisfying Fn+k(m)≡Fn( mod m),F_{n+k(m)}\equiv F_n(\bmod m), for all natural numbers nn. a) Prove that: For all m1,m2∈Z+m_1,m_2\in \mathbb{Z^+}, we have:k([m1,m2])=[k(m1),k(m2)].k([m_1,m_2])=[k(m_1),k(m_2)].(Here [a,b][a,b] is the least common multiple of a,b.a,b.)
b) Determine k(2),k(4),k(5),k(10).k(2),k(4),k(5),k(10).
algebra