Algebra
# Absolute Value Inequalities

For all ordered triples of real values \( (a, b, c) \), which of these numbers is larger?

If \( x, y, z \) are non-zero real, what is the range of

\[ f(x, y, z) = \frac{ |x+y| } { |x| + |y| } + \frac{ |y+z| } { |y| + |z| } + \frac{ |z+x| } { |z| + |x| } ? \]

Consider all monic polynomials \(f(x) = x^2 + bx + c \), where \(b\) and \(c\) are real numbers. What is the minimum value of \(N\), where

\[ N = \max_{x \in [-22, 50]} \vert f(x) \vert? \]

Consider all monic polynomials of the form \(f_a(x) = x + a \). As \(a\) ranges over all real numbers, what is the minimum value of \(N_a\), where

\[ N_a = \max_{x \in [-13, 39] } \lvert f_a(x) \rvert ? \]

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