Algebra › Polynomials

Chains of quadratics

Solve the first problem 3 problems

1. Maximum number of quadratic equations on a board ●●●○○ 50 ptsWhat is the maximum number of different monic quadratic equations that can be written on a board, given that any two of them have a common root, but no four of them have a root common to all?
2. Counting super keys from dependencies ●●●○○ 50 ptsA relation $R$ has attributes $A, B, C, D, E, F, G, H$ and the functional dependencies $CH \to G$, $A \to BC$, $B \to CFH$, $E \to A$, $F \to EG$. How many super keys does $R$ have? A super key is a set of attributes whose closure contains all attributes of $R$.
3. Longest chain of quadratic equations ●●●●○ 100 ptsQuadratic equations are solved one after another. Each equation has leading coefficient 1, of the form $x^2+p_jx+q_j=0$. Whenever an equation has two distinct real roots, the next equation is formed as follows: its constant term is the larger root of the current equation, and its linear coefficient is the smaller root of the current equation. The process stops at the first equation that does not have two distinct real roots. Find the maximum possible number of quadratic equations in such a sequence.

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