Problem
ALG-B1-M03-P026 Must All Numbers Be Equal?
Positive numbers \(x,y,z\) have the following property: the values \(x+2y^2+2z^2\), \(y+2z^2+2x^2\), \(z+2x^2+2y^2\) are equal. Must \(x=y=z\)?
Subtract two equal expressions. A factor \(x-y\) should appear.
Answer: no, not necessarily. We show how to find a counterexample.
From the equality of the first two expressions, \(x+2y^2=y+2x^2\). Move all terms to one side:
\[x-y-2(x^2-y^2)=0.\]
Thus \((x-y)(1-2(x+y))=0\). Therefore for each pair of variables, either they are equal or their sum is \(\frac12\).
Take \(x=y=\frac16\), \(z=\frac13\). Then \(x+y=\frac13\), but \(x+z=y+z=\frac12\). Check:
\[x+2y^2+2z^2=\frac16+\frac1{18}+\frac2{9}=\frac49,\]
and the other two expressions are also equal to \(\frac49\). However, \(x,y,z\) are not all equal. Hence the statement is false.
Rewritten regional idea: equality of similar expressions leads to a factor \((x-y)\) and the alternative “equal or fixed sum”.