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I have this problem from USAMO 2002,

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

For which I found a solution here

https://artofproblemsolving.com/wiki/index.php?title=2002_USAMO_Problems/Problem_3

However I have a doubt in the last step of the solution, How is the author concluding that P(x) has n real roots, I followed the solution perfectly till the step where he proves that P(x) has at least n - 1 real roots?

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