Algebra · Absolute value · Piecewise linear functions · Counting solutions · Extremal construction
Problem 4, 2024
← Prev · 18 / 18 · Next →Determine the smallest positive integer \(n\) with the following property: for some real numbers \(a_0, a_1, \dots, a_n\) the function \(f : \mathbb{R} \to \mathbb{R}\) defined by
\[ f(x) = \bigl| \, \cdots \, \bigl| \bigl| \, |x - a_0| - a_1 \bigr| - a_2 \bigr| - \cdots - a_{n-1} \bigr| - a_n, \qquad x \in \mathbb{R}, \]
has exactly \(2023\) real zeros. (A point \(x_0 \in \mathbb{R}\) is a zero of \(f\) when \(f(x_0) = 0\).)
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Serbian National Competition (Drzavno takmicenje) 2024, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source