2020 AMC 10B Problems/Problem 24
How many positive integers satisfy(Recall that is the greatest integer not exceeding .)
(Quick solution if you’re in a hurry)
First notice that the graphs of and intersect at 2 points. Then, notice that must be an integer. This means that n is congruent to .
For the first intersection, testing the first few values of (adding to 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 .
Solution 2 (Graphing)
One intuitive approach to the question is graphing. Obviously, you should know what the graph of the square root function is, and if any function is floored (meaning it is taken to the greatest integer less than a value), a stair-like figure should appear. The other function is simply a line with a slope of . If you precisely draw out the two regions of the graph where the derivative of the square function nears the derivative of the linear function, you can now deduce that values of intersection lay closer to the left side of the stair, and values lay closer to the right side of the stair.
With meticulous graphing, you can realize that the answer is .
A in-depth graph with intersection points is linked below. https://www.desmos.com/calculator/e5wk9adbuk
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