Problem 1

Solve

x+y+z=a,\quad x^2+y^2+z^2=b^2,\quad xy=z^2

in real numbers. Find the necessary and sufficient condition on

a

and

b

for

x,y,z

to be distinct positive numbers.

Step 3 of 3: Recover x,y and the positivity condition

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In plain words

For each admissible z, x and y are the two roots of one quadratic; its discriminant controls reality and distinctness.

x,y=\frac{a-z\pm\sqrt{a^2-2az-3z^2}}{2},\qquad a^2-2az-3z^2\ge0

Detailed analysis

For each z from Step 2, x and y are the roots of $t^2-(a-z)t+z^2=0$ , giving the displayed formula; the second inequality is exactly the condition for real x,y. This parametrizes all real solutions (including the zero solution when $a=b=0$ ). If x,y,z must instead be distinct positive numbers, positivity gives $a>0$ and $|b|<a$ , while distinctness adds $3b^2-a^2>0$ . Hence the necessary and sufficient condition is $a>0$ and $a/\sqrt3<|b|<a$ .