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x^2 + p_1x + q_1 = 0, x^2 + p_2x + q_2 = 0, x^2 + p_3x + q_3 = 0,

Source: Polish MO Second Round 1960 p2

August 31, 2024
algebrasystem of equations

Problem Statement

The equations are given x2+p1x+q1=0x2+p2x+q2=0x2+p3x+q3=0 \begin{array}{c} x^2 + p_1x + q_1 = 0\\ x^2 + p_2x + q_2 = 0\\ x^2 + p_3x + q_3 = 0 \end{array} each two of which have a common root, but all three have no common root. Prove that:
1) 2(p1p2+p2p3+p3p1)−(p12+p22+p32)=4(q1+q2+q3)2 (p_1p_2 + p_2p_3 + p_3p_1) - (p_1^2 + p_2^2 + p_3^2) = 4 (q_1 + q_2+ q_3)
2) he roots of these equations are rational when the numbers p1p_1, p2p_2 and p3p_3 are rational}.
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