<p>Let <em>a</em> > 2 be a constant. If there are just 18 positive integers satisfying the inequality \((x - a)(x - 2a)(x - a^2) < 0\), then find the value of <em>a</em>.</p>
Step-by-Step Solution
Key Concept: The product (x - a)(x - 2a)(x - a²) < 0 changes sign at three critical points; for a > 2, we have a < 2a < a², so the solution set is the union of two intervals where the product is negative. Count positive integers in these intervals to equal 18.
<p><strong>Step 1: Identify critical points.</strong> Since a > 2, we have: a < 2a < a² (ordering is correct since 2a > a when a > 2, and a² > 2a when a > 2).</p><p><strong>Step 2: Determine sign pattern.</strong> For the product (x - a)(x - 2a)(x - a²):</p><ul><li>When x < a: all three factors negative → product NEGATIVE ✓</li><li>When a < x < 2a: first factor positive, other two negative → product POSITIVE</li><li>When 2a < x < a²: first two positive, third negative → product NEGATIVE ✓</li><li>When x > a²: all three positive → product POSITIVE</li></ul><p><strong>Step 3: Solution set.</strong> The inequality (x - a)(x - 2a)(x - a²) < 0 is satisfied when x ∈ (−∞, a) ∪ (2a, a²).</p><p><strong>Step 4: Count positive integers.</strong> Positive integers in (−∞, a) are: {1, 2, 3, ..., ⌊a⌋} if a is not an integer, giving ⌊a⌋ integers (or ⌊a⌋ - 1 if a is an integer). Since a > 2, assume a is not an integer: this gives ⌊a⌋ positive integers.</p><p>Positive integers in (2a, a²) are those n where 2a < n < a². The count is ⌊a²⌋ − ⌊2a⌋.</p><p><strong>Step 5: Set up equation.</strong> Total positive integers: ⌊a⌋ + (⌊a²⌋ − ⌊2a⌋) = 18.</p><p><strong>Step 6: Test a = 5.</strong> Then a = 5, 2a = 10, a² = 25.</p><ul><li>Positive integers in (−∞, 5): {1, 2, 3, 4} → 4 integers</li><li>Positive integers in (10, 25): {11, 12, ..., 24} → 14 integers</li><li>Total: 4 + 14 = 18 ✓</li></ul><p>∴ <strong>a = 5</strong></p>
Correct Answer: 5