MathDB

Problem 7

Part of 2022 VN Math Olympiad For High School Students

Problems(2)

Fermat point, orthocenter

Source: 2022 Viet Nam math olympiad for high school students D1 P7

3/20/2023
Let ABCABC be a triangle with ∠A,∠B,∠C<120∘\angle A,\angle B,\angle C <120^{\circ}, TT is its Fermat-Torricelli point. Let Ha,Hb,HcH_a, H_b, H_c be the orthocenter of triangles TBC,TCA,TABTBC, TCA, TAB, respectively. a) Prove that: TT is the centroid of the △HaHbHc\triangle H_aH_bH_c. b) Denote D,E,FD, E, F respectively by the intersections of HcHbH_cH_b and the segment BCBC, HcHaH_cH_a and the segment CACA, HaHbH_aH_b and the segment ABAB. Prove that: the triangle DEFDEF is equilateral. c) Prove that: the lines passing through D,E,FD, E, F and are respectively perpendicular to BC,CA,ABBC, CA, AB are concurrent at a point. Let that point be SS. d) Prove that: TSTS is parallel to the Euler line of the triangle ABCABC.
geometry
Fibonacci sequence, congruent, k(2^s)

Source: 2022 Viet Nam math olympiad for high school students D2 P7

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, ss is a positive integer. Prove that: a) F3.2s−1≡0( mod 2s){F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s}) and F3.2s−1+1≡1( mod 2s).{F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}). b) k(2s)=3.2s−1.k({2^s}) = {3.2^{s - 1}}.
algebranumber theory