Difference between revisions of "2007 AMC 12B Problems/Problem 18"
Chickendude (talk | contribs) (New page: ==Problem 18== Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. The three-digit integer <math>abc</math> lies one third of the way from the square...) |
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Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. The three-digit integer <math>abc</math> lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer <math>acb</math> lies two thirds of the way between the same two squares. What is <math>a+b+c</math>? | Let <math>a</math>, <math>b</math>, and <math>c</math> be digits with <math>a\ne 0</math>. The three-digit integer <math>abc</math> lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer <math>acb</math> lies two thirds of the way between the same two squares. What is <math>a+b+c</math>? | ||
| − | <math>\mathrm {(A)} 10 | + | <math>\mathrm{(A)}\ 10 \qquad \mathrm{(B)}\ 13 \qquad \mathrm{(C)}\ 16 \qquad \mathrm{(D)}\ 18 \qquad \mathrm{(E)}\ 21</math> |
==Solution== | ==Solution== | ||
Revision as of 00:05, 21 February 2008
Problem 18
Let 
, 
, and 
be digits with 
. The three-digit integer 
lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer 
lies two thirds of the way between the same two squares. What is 
?
Solution
The difference between and is given by
The difference between the two squares is three times this amount or
The difference between two consecutive squares is always an odd number. The consecutive squares with common difference 
are 
and 
. One third of the way between them is 
and two thirds of the way is 
This gives 
, 
, 
