Math Olympiad Algebra Challenge | Poland Math Olympiad Problem

Math Olympiad Algebra Challenge | Poland Math Olympiad Problem

[lifeisajourney]

Hello, good problem
I don't know why this works
a+2=b+c-2
b+4=c+a-4
c+6=a+b-6
add all the equations
a+b+c+12=2a+2b+2c-12
a+b+c=24

[dlnkbfv]

I knew Titu's inequality for the first time. This is one of applications of Cauchy-Schwartz inequality.
For every real number x, y, z and real positive number p, q, r we have
( x + y + z )^2 = ( √p・x/√p + √q・y/√q + √r・z/√r )^2
<= ( p + q + r )( x^2/p + y^2/q + z^2/r )
( x + y + z )^2/( p + q + r ) <= x^2/p + y^2/q + z^2/r
The equality holds when x/√p : y/√q : z/√r = √p : √q : √r .
This is equivalent to x/p = y/q = z/r .