Difference between revisions of "2020 AMC 10B Problems/Problem 24"
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First notice that the graphs of <math>(x+1000)/70</math> and <math>\sqrt[]{n}</math> intersect at 2 points. Then, notice that <math>(n+1000)/70</math> must be an integer. This means that n is congruent to 50 (mod 70). | First notice that the graphs of <math>(x+1000)/70</math> and <math>\sqrt[]{n}</math> intersect at 2 points. Then, notice that <math>(n+1000)/70</math> must be an integer. This means that n is congruent to 50 (mod 70). | ||
| − | For the first intersection, testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields <math>n=20</math> and <math>n=21</math>. | + | For the first intersection, testing the first few values of <math>n</math> (adding <math>70</math> to n each time and noticing the left side increases by <math>1</math> each time) yields <math>n=20</math> and <math>n=21</math>. |
For the second intersection, using binary search can narrow down the other cases, being <math>n=47</math>, <math>n=48</math>, <math>n=49</math>, and <math>n=50</math>. This results in a total of 6 cases, for an answer of <math>\boxed{\textbf{(C) }6}</math>. | For the second intersection, using binary search can narrow down the other cases, being <math>n=47</math>, <math>n=48</math>, <math>n=49</math>, and <math>n=50</math>. This results in a total of 6 cases, for an answer of <math>\boxed{\textbf{(C) }6}</math>. | ||
Revision as of 12:16, 8 February 2020
Problem
How many positive integers 
satisfy![\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\]](http://latex.artofproblemsolving.com/0/d/d/0dd28ce1709d2e91e3b6ff963c8adf1cac05b9fa.png)
(Recall that 
is the greatest integer not exceeding 
.)
Solution
(Quick solution if you’re in a hurry)
First notice that the graphs of 
and ![$\sqrt[]{n}$](http://latex.artofproblemsolving.com/b/f/1/bf1c18f78bd8f62980b31821dd7585c37733fed4.png)
intersect at 2 points. Then, notice that 
must be an integer. This means that n is congruent to 50 (mod 70).
For the first intersection, testing the first few values of (adding 
to n each time and noticing the left side increases by 
each time) yields 
and 
.
For the second intersection, using binary search can narrow down the other cases, being 
, 
, 
, and 
. This results in a total of 6 cases, for an answer of 
.
~DrJoyo
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
