Difference between revisions of "2020 AMC 10B Problems/Problem 24"
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| − | First notice that (n+1000)/70 must be an integer. This means that n is congruent to 50 (mod 70). Testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields n=20 and n=21. At this point, the right side increases faster than the left. However, since the left side is linear and the right side is roughly a radical, the graphs must intersect in at least one other point. Using binary search can narrow down the other cases, being n=47, n=48, n=49, and n=50. | + | First notice that (n+1000)/70 must be an integer. This means that n is congruent to 50 (mod 70). Testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields n=20 and n=21. At this point, the right side increases faster than the left. |
| + | However, since the left side is linear and the right side is roughly a radical, the graphs must intersect in at least one other point. Using binary search can narrow down the other cases, being n=47, n=48, n=49, and n=50. | ||
~DrJoyo | ~DrJoyo | ||
Revision as of 02:32, 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 (n+1000)/70 must be an integer. This means that n is congruent to 50 (mod 70). Testing the first few values of n (adding 70 to n each time and noticing the left side increases by 1 each time) yields n=20 and n=21. At this point, the right side increases faster than the left. However, since the left side is linear and the right side is roughly a radical, the graphs must intersect in at least one other point. Using binary search can narrow down the other cases, being n=47, n=48, n=49, and n=50. ~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 | ||
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