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.  
 
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.
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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. This is 6 cases, for an answer of C.
  
 
~DrJoyo
 
~DrJoyo

Revision as of 03:33, 8 February 2020

Problem

How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)

$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

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. This is 6 cases, for an answer of C.

~DrJoyo

See Also

2020 AMC 10B (ProblemsAnswer KeyResources)
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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