MathDB

Problem 3

Part of 2007 Mongolian Mathematical Olympiad

Problems(2)

reflections to get point inside triangle

Source: Mongolian MO 2007 Grade 11 P3

4/8/2021
Let PP be a point outside of the triangle ABCABC in the plane of ABCABC. Prove that by using reflections SABS_{AB}, SACS_{AC}, and SBCS_{BC} across the lines ABAB, ACAC, and BCBC one can shift point PP inside the triangle ABCABC.
geometrygeometric transformationreflection
equal sets mod p

Source: Mongolian MO 2007 Teachers P3

4/8/2021
Let pp be an odd prime number. Let gg be a primitive root of unity modulo pp. Find all the values of pp such that the sets A={k2+1:1≤k≤p−12}A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\} and B={gm:1≤m≤p−12}B=\left\{g^m:1\le m\le\frac{p-1}2\right\} are equal modulo pp.
number theory