Problem
ALG-B3-M02-P021 One axis for three parabolas
Three quadratic polynomials \(A(x),B(x),C(x)\) have positive leading coefficients and each has two distinct real roots. If \(A(x)+B(x)\) has equal values at the two roots of \(C\), \(B(x)+C(x)\) has equal values at the two roots of \(A\), and \(C(x)+A(x)\) has equal values at the two roots of \(B\), prove that the sums of the roots of the three polynomials are equal.
Hint 1. Consider \(S(x)=A(x)+B(x)+C(x)\).
Hint 2. A quadratic function takes equal values at two points symmetric about its axis.
Let \(a_1,a_2\), \(b_1,b_2\), \(c_1,c_2\) be the roots of \(A,B,C\). Consider \(S=A+B+C\). At \(c_1,c_2\), \(C(c_1)=C(c_2)=0\), so equal values of \(A+B\) give \(S(c_1)=S(c_2)\). Similarly, \(S(a_1)=S(a_2)\) and \(S(b_1)=S(b_2)\). Since \(S\) is a quadratic polynomial with positive leading coefficient, two distinct points with equal values are symmetric about the same axis \(x=d\). Therefore \(a_1+a_2=b_1+b_2=c_1+c_2=2d\).
A. Source analysis. Main objects: three quadratic functions and equal values at the roots of another function.
B. Insufficient first move. Working with each pair separately does not connect the three axes.
C. Hidden observation. The sum of the three polynomials has the same equal-value property at each pair of roots.
D. Required move. Introduce S=A+B+C and use symmetry about the axis of a parabola.
E. Number of ideas. Two ideas: substituting roots and finding one common axis.
F. Difficulty justification. Regional level 6: the solution is short but requires combining three conditions into one function.
G. Why it is not one-step. It is not one-step: without S, the conditions look independent.