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Part of 2003 Putnam

Problems(2)

Putnam 2003 A4

Source:

6/23/2011
Suppose that a,b,c,A,B,Ca, b, c, A, B, C are real numbers, a≠0a \not= 0 and A≠0A \not= 0, such that ∣ax2+bx+c∣≤∣Ax2+Bx+C∣|ax^2+ bx + c| \le |Ax^2+ Bx + C| for all real numbers xx. Show that ∣b2−4ac∣≤∣B2−4AC∣|b^2- 4ac| \le |B^2- 4AC|
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Putnam 2003 B4

Source:

6/23/2011
Let f(z)=az4+bz3+cz2+dz+e=a(z−r1)(z−r2)(z−r3)(z−r4)f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4) where a,b,c,d,ea, b, c, d, e are integers, a≠0a \not= 0. Show that if r1+r2r_1 + r_2 is a rational number, and if r1+r2≠r3+r4r_1 + r_2 \neq r_3 + r_4, then r1r2r_1r_2 is a rational number.
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