An algebra problem by Varun Agrawal
Algebra
Level
3
Find x+y+z for integer solutions for
x^3= 2y-1
y^3= 2z-1
z^3= 2x-1
The answer is 3.
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1 solution
ASSUME X>Y.
THEN X^3>Y^3.
THEN 2Y-1>2Z-1.
THEREFORE Y>Z.
NOW Y^3>Z^3.
SO 2Z-1>2X-1.
HENCE Z>X.
WE GET X>Y>Z>X. THIS IS IMPOSSIBLE. THEREFORE X=Y=Z.
SUBSTITUTING , X^3=2X-1.
X^3-2X+1=0
(X-1)(X^2 +X - 1)=0
THE ONLY INTEGER SOLUTION IS X=Y=Z=1.
THEREFORE X+Y+Z=3.