MathDB

Problem 5

Part of 2022 VN Math Olympiad For High School Students

Problems(2)

Steiner's problem

Source: 2022 Viet Nam math olympiad for high school students D1 P5

3/20/2023
Given a convex quadrilateral MNPQMNPQ. Assume that there exists 2 points U,VU, V inside MNPQMNPQ satifying:∠MUN=∠MUV=∠NUV=∠QVU=∠PVU=∠PVQ\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQConsider another 2 points X,YX, Y in the plane. Prove that the sumXM+XN+XY+YP+YQXM + XN + XY + YP + YQget its minimum value iff X≡U,Y≡VX\equiv U, Y\equiv V.
geometry
Fibonacci sequence, congruent mod p

Source: 2022 Viet Nam math olympiad for high school students D2 P5

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, pp is an odd prime such that p≡±1( mod 5)p \equiv \pm 1(\bmod 5). Prove that: a) 5p−12≡1( mod p).{5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p). b) Fp−1≡0( mod p).{F_{p - 1}} \equiv 0(\bmod p). c) k(p)∣p−1.k(p)|p-1.
algebranumber theory