parmenides51 29.08.2019 00:24 Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.
Semcio 31.08.2019 23:45 Actually it is also reversed Putnam 2003 https://mks.mff.cuni.cz/kalva/putnam/putn03.html
RayThroughSpace 01.09.2019 04:57 $(x_1 + x_2) + (x_3 + x_4) \in \mathbb{Q} \implies (x_1 + x_2) \in \mathbb{Q}$ $(x_1 + x_2)(x_3+x_4) + x_3x_4 + x_1x_2 \in \mathbb{Q} \implies x_3x_4 +x_1x_2 \in \mathbb{Q} ... (1)$ $x_1x_2x_3x_4 \in \mathbb{Q} \implies x_1x_2 \in \mathbb{R-Q}$ We also have from $S_3$, $x_3x_4 + x_1x_2 (x_3 +x_4)/(x_1 + x_2) \in \mathbb{Q} ...(2)$. $(2) - (1)$ gives $(x_3 + x_4)/(x_1 + x_2) = 1$