Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2007 AMC 12B Problems/Problem 17"

m (→‎Solution)
(→‎Solution)
 
Line 6: Line 6:
 
==Solution==
 
==Solution==
  
−
Note that if <math>a</math> is positive, then, the equation will have no solutions for <math>b</math>. This becomes more obvious by noting that at <math>b=1</math>, <math>ab^2 > \log_{10} b</math>. The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.
+
Note that if <math>a</math> is positive, then, the equation will have no solutions for <math>b</math>. This becomes more obvious by noting that at <math>a=1</math>, <math>ab^2 > \log_{10} b</math>. The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.
  
 
This puts <math>a</math> as the smallest in the set since it must be negative.
 
This puts <math>a</math> as the smallest in the set since it must be negative.
Line 12: Line 12:
 
Checking the new equation: <math>-b^2 = \log_{10}b</math>
 
Checking the new equation: <math>-b^2 = \log_{10}b</math>
  
−
Near <math>b=0</math>, <math>-b^2 > \log_{10} b</math> but at <math>a=1</math>, <math>-b^2 < \log_{10} b</math>
+
Near <math>b=0</math>, <math>-b^2 > \log_{10} b</math> but at <math>b=1</math>, <math>-b^2 < \log_{10} b</math>
  
 
This implies that the solution occurs somewhere in between: <math>0 < b < 1</math>
 
This implies that the solution occurs somewhere in between: <math>0 < b < 1</math>

Latest revision as of 02:05, 19 May 2024

Problem

If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b$, what is the median of the set $\{0,1,a,b,1/b\}$?

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ a \qquad \mathrm{(D)}\ b \qquad \mathrm{(E)}\ \frac{1}{b}$

Solution

Note that if $a$ is positive, then, the equation will have no solutions for $b$. This becomes more obvious by noting that at $a=1$, $ab^2 > \log_{10} b$. The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.

This puts $a$ as the smallest in the set since it must be negative.

Checking the new equation: $-b^2 = \log_{10}b$

Near $b=0$, $-b^2 > \log_{10} b$ but at $b=1$, $-b^2 < \log_{10} b$

This implies that the solution occurs somewhere in between: $0 < b < 1$

This also implies that $\frac{1}{b} > 1$

This makes our set (ordered) $\{a,0,b,1,1/b\}$

The median is $b \Rightarrow \mathrm {(D)}$

Solution 2

Let $b=0.1$. Then $a\cdot0.01 = -1,$ giving $a=-100$. Then the ordered set is $\{-100, 0, 0.1, 1, 10\}$ and the median is $0.1=b,$ so the answer is $\mathrm {(D)}$.

See Also

2007 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_AMC_12B_Problems/Problem_17&oldid=219485"