MathDB

Problem 1

Part of 2022 VN Math Olympiad For High School Students

Problems(2)

only 1 Fermat-Torricelli point inside a triangle

Source: 2022 Viet Nam math olympiad for high school students D1 P1

3/20/2023
Let ABCABC be a triangle with ∠A,∠B,∠C<120∘\angle A,\angle B,\angle C <120^{\circ}.
Prove that: there is exactly one point TT inside △ABC\triangle ABC such that ∠BTC=∠CTA=∠ATB=120∘\angle BTC=\angle CTA=\angle ATB=120^{\circ}. (TT is called Fermat-Torricelli point of △ABC\triangle ABC)
geometry
basic Fibonacci sequence properties

Source: 2022 Viet Nam math olympiad for high school students D2 P1

3/21/2023
Given Fibonacci sequence (Fn)(F_n) a) Prove that: for all u,v∈N,u≥1u,v\in \mathbb{N}, u\ge 1, we have:Fu+v=Fu−1Fv+FuFv+1.F_{u+v}=F_{u-1}F_{v}+F_{u}F_{v+1}. b) Prove that: for all n∈N,n≥1n\in \mathbb{N}, n\ge 1, we have:F2n=Fn(Fn−1+Fn+1),F_{2n}=F_n(F_{n-1}+F_{n+1}),F2n+1=Fn2+Fn+12.F_{2n+1}=F_n^2+F_{n+1}^2.
algebra