MathDB

2

Part of 2012 Czech-Polish-Slovak Junior Match

Problems(2)

a^2+ab+b^2=c^2+3 prime diophantine

Source: Czech-Polish-Slovak Junior Match 2012, Individual p2 CPSJ

3/11/2020
Determine all three primes (a,b,c)(a, b, c) that satisfied the equality a2+ab+b2=c2+3a^2+ab+b^2=c^2+3.
Diophantine equationdiophantineprimesnumber theory
parallel and locus of midpoints wanted, as a point moves along a long arc

Source: Czech-Polish-Slovak Junior Match 2012, Team p2 CPSJ

3/12/2020
On the circle kk, the points A,BA,B are given, while ABAB is not the diameter of the circle kk. Point CC moves along the long arc ABAB of circle kk so that the triangle ABCABC is acute. Let D,ED,E be the feet of the altitudes from A,BA, B respectively. Let FF be the projection of point DD on line ACAC and GG be the projection of point EE on line BCBC. (a) Prove that the lines ABAB and FGFG are parallel. (b) Determine the set of midpoints SS of segment FGFG while along all allowable positions of point CC.
midpointparallelgeometryLocusarc