Let P1(x) = x^2 + a1x + b1 and P2(x) = x^2 + a2x + b2 be two quadratic polynomials with integer coeffcients.
Let P1(x) = x2 + a1x + b1 and P2(x) = x2 + a2x + b2 be two quadratic polynomials with integer coeffcients. Suppose a1 ≠ a2 and there exist integers m ≠ n such that P1(m) = P2(n), P2(m) = P1(n). Prove that a1 - a2 is even.
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We have
m2 + a1m + b1 = n2 + a2n + b2
n2 + a1n + b1 = m2 + a2m + b2.
Hence
(a1 - a2)(m + n) = 2(b2 - b1); (a1 + a2)(m - n) = 2(n2 - m2).
This shows that a1 + a2 = -2(n + m).
Hence
4(b2 - b1) = a22 - a21.
Since a1 + a2 and a1 - a2 have same parity, it follows that a1 - a2 is even.
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