<s>[INST] Winning strategy of Peter such that quadratic equations has Rational roots Peter picked three non zero real numbers $a,b,c$  and Alan arranges these numbers as coefficients of Quadratic equation $ax^2+bx+c=0$.
Peter wins if this Quadratic has distinct Rational roots, else Alan wins. 
How can we prove that Peter always has winning strategy.
Since $a,b,c$ are non zero reals we have roots of quadratic as
$$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
Now Peter can win game only if $$b^2-4ac=k^2$$ where $k$ is Integer. But even though $b^2-4ac=k^2$ is integer, how can we say roots are rational?
 [/INST] It's enough to find one set of real numbers that Peter can choose to win. For example, $\{1, 2, -3\}$ will work. We can check that:
\begin{align}
  x^2 + 2x - 3 &= (x+3)(x-1) \\
  x^2 - 3x + 2 &= (x-2)(x-1) \\
  2x^2 + x - 3 &= (2x+3)(x-1) \\
  2x^2 - 3x + 1 &= (2x-1)(x-1) \\
  -3x^2 + x + 2 &= -(3x+2)(x-1) \\
  -3x^2 + 2x + 1 &= -(3x+1)(x-1)
\end{align}
and so both roots of all six polynomials Alan can form are rational and distinct.

I found this by brute force, but in retrospect it seems obvious why this worked and how we can find lots more such triples. Since $1+2+(-3)=0$, $x=1$ will always be a root of the quadratic equation, and factoring out $x-1$ must produce a second rational root.
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