MathDB

N2

Part of 2016 Costa Rica - Final Round

Problems(2)

a^4 + 4b^4 is prime

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist N2 day1

9/23/2021
Determine all positive integers aa and bb for which a4+4b4a^4 + 4b^4 be a prime number.
primenumber theory
x + y + z divides x^2 + y^2 + z^2 when x^3=y^3=z^3 same remainder mod p

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist N2 day2

9/24/2021
Let x,y,zx, y, z be positive integers and pp a prime such that x<y<z<px <y <z <p. Also x3,y3,z3x^3, y^3, z^3 leave the same remainder when divided by pp. Prove that x+y+zx + y + z divides x2+y2+z2x^2 + y^2 + z^2.
dividesdivisiblenumber theory