MathDB

Problem 8

Part of 2022 VN Math Olympiad For High School Students

Problems(2)

AP is perpendicular to QR iff AB=AC or 2BC^2=AB^2+AC^2

Source: 2022 Viet Nam math olympiad for high school students D1 P8

3/20/2023
Given the triangle ABCABC with TT is its Fermat–Torricelli point. Let (Na)(N_a) be the circumcircle of △TBC\triangle TBC. Choose a point XX on (Na)(N_a) such that TXTX is perpendicular to BCBC. The segment BCBC intersects (TNaX)(TN_aX) at DD. Similar definition of points Y,Z,E,FY, Z, E, F. The reflection lines of the Euler line of △ABC\triangle ABC wrt BC,CA,ABBC, CA, AB intersect XD,YE,ZFXD, YE, ZF at P,Q,RP, Q, R, respectively.
Prove that: APAP is perpendicular to QRQR if and only if AB=ACAB = AC or 2BC2=AB2+AC22BC^2 = AB^2 + AC^2.
k(m) is even

Source: 2022 Viet Nam math olympiad for high school students D2 P8

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.
Prove that: k(m)k(m) is even for all m>2.m>2.
algebranumber theory