MathDB

5

Part of 2023 New Zealand MO

Problems(2)

a+b =n/2, and a,b both divisors of n

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p5

9/2/2023
Find all triples (a,b,n)(a, b, n) of positive integers such that aa and bb are both divisors of nn, and a+b=n2a+b = \frac{n}{2} .
number theory
x + y + z is integer if x^2 = y + 2 and y^2 = z + 2 and z^2 = x + 2

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p5

9/2/2023
Let x,yx, y and zz be real numbers such that: x2=y+2x^2 = y + 2, and y2=z+2y^2 = z + 2, and z2=x+2z^2 = x + 2. Prove that x+y+zx + y + z is an integer.
algebrasystem of equations