Problem
ALG-B2-M01-P009 Three quadratic trinomials
Let \(F_i(x)=x^2+2p_i x+q_i\), \(i=1,2,3\). Suppose \(p_1p_2p_3=q_1q_2q_3=N>1\). Prove that at least one trinomial has two distinct real roots.
Hint 1. Assume all discriminants are nonpositive.
Hint 2. Multiplication is valid only after checking the signs of \(q_i\).
Assume the contrary. Then for each \(i\), the discriminant \(4(p_i^2-q_i)\le0\), so \(p_i^2\le q_i\). Hence all \(q_i\ge0\), and the inequalities may be multiplied: \((p_1p_2p_3)^2\le q_1q_2q_3\). Thus \(N^2\le N\), impossible for \(N>1\). Therefore for some \(i\), \(p_i^2>q_i\), so the corresponding trinomial has two distinct real roots.
A. Source analysis. Main objects: inequalities, order, an extremal element, or an invariant. The obvious first move usually gives only a local estimate. The hidden observation is to choose the right nondecreasing quantity, or to add/multiply inequalities only after signs are controlled. The needed step is an ordering, an invariant, a product transformation, or a boundary case.
F. Difficulty justification. Regional level 6: one hidden step is to multiply discriminant estimates only after signs are checked.
G. Why this is not a one-step exercise. One must not only recall the discriminant, but connect three trinomials through the product.