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Difference between revisions of "1981 AHSME Problems/Problem 28"

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==Problem 28==
 
==Problem 28==
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Consider the set of all equations <math> x^3 + a_2x^2 + a_1x + a_0 = 0</math>, where <math> a_2</math>, <math> a_1</math>, <math> a_0</math> are real constants and <math> |a_i| < 2</math> for <math> i = 0,1,2</math>. Let <math> r</math> be the largest positive real number which satisfies at least one of these equations. Let <math> r</math> be the largest positive real number that satisfies at least one of these equations. Which of the inequalities below does <math> r</math> satisfy?
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Consider the set of all equations <math> x^3 + a_2x^2 + a_1x + a_0 = 0</math>, where <math> a_2</math>, <math> a_1</math>, <math> a_0</math> are real constants and <math> |a_i| \leq 2</math> for <math> i = 0,1,2</math>. Let <math>r</math> be the largest positive real number which satisfies at least one of these equations. Then
  
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<math> \textbf{(A)}\ 1 < r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \dfrac{7}{2}</math>
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<math> \textbf{(A)}\ 1 \leq r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} \leq r < 2\qquad \textbf{(C)}\ 2 \leq r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} \leq r < 3\qquad\ \textbf{(E)}\ 3 \leq r < \dfrac{7}{2}</math>
  
 
==Solution==
 
==Solution==

Latest revision as of 20:41, 19 August 2025

Problem 28

Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| \leq 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then

$\textbf{(A)}\ 1 \leq r < \dfrac{3}{2}\qquad \textbf{(B)}\ \dfrac{3}{2} \leq r < 2\qquad \textbf{(C)}\ 2 \leq r < \dfrac{5}{2}\qquad \textbf{(D)}\ \dfrac{5}{2} \leq r < 3\qquad\ \textbf{(E)}\ 3 \leq r < \dfrac{7}{2}$

Solution

Since $x^3 = -(a_2x^2 + a_1x + a_0)$ and $x$ will be as big as possible, we need $x^3$ to be as big as possible, which means $a_2x^2 + a_1x + a_0$ is as small as possible. Since $x$ is positive (according to the options), it makes sense for all of the coefficients to be $-2$.

Evaluating $f(\frac{5}{2})$ gives a negative number, $f(3)$ 1, and $f(\frac{7}{2})$ a number greater than 1, so the answer is $\boxed{D}$

See also

1981 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 27 		Followed by
Problem 29
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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