Difference between revisions of "2018 AMC 10B Problems/Problem 25"

Revision as of 00:55, 31 July 2025

The following problem is from both the 2018 AMC 10B #25 and 2018 AMC 12B #24, so both problems redirect to this page.

Problem

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ ?

$\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$

Solution 1

This rewrites itself to $x^2=10,000\{x\}$ where $\lfloor x \rfloor + \{x\} = x$ .

Graphing $y=10,000\{x\}$ and $y=x^2$ we see that the former is a set of line segments with slope $10,000$ from $0$ to $1$ with a hole at $x=1$ , then $1$ to $2$ with a hole at $x=2$ etc. Here is a graph of $y=x^2$ and $y=16\{x\}$ for visualization.

$[asy] import graph; size(400); xaxis("$x$",Ticks(Label(fontsize(8pt)),new real[]{-5,-4,-3, -2, -1,0,1 2,3, 4,5})); yaxis("$y$",Ticks(Label(fontsize(8pt)),new real[]{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18})); real y(real x) {return x^2;} draw(circle((-4,16), 0.1)); draw(circle((-3,16), 0.1)); draw(circle((-2,16), 0.1)); draw(circle((-1,16), 0.1)); draw(circle((0,16), 0.1)); draw(circle((1,16), 0.1)); draw(circle((2,16), 0.1)); draw(circle((3,16), 0.1)); draw(circle((4,16), 0.1)); draw((-5,0)--(-4,16), black); draw((-4,0)--(-3,16), black); draw((-3,0)--(-2,16), black); draw((-2,0)--(-1,16), black); draw((-1,0)--(-0,16), black); draw((0,0)--(1,16), black); draw((1,0)--(2,16), black); draw((2,0)--(3,16), black); draw((3,0)--(4,16), black); draw(graph(y,-4.2,4.2),green); [/asy]$

Now notice that when $x=\pm 100$ the graph has a hole at $(\pm 100,10,000)$ which the equation $y=x^2$ passes through and then continues upwards. Thus our set of possible solutions is bounded by $(-100,100)$ . We can see that $y=x^2$ intersects each of the lines once and there are $99-(-99)+1=199$ lines for an answer of $\boxed{\text{(C)}~199}$ .

Solution 2

Same as the first solution, $x^2=10,000\{x\}$ .

We can write $x$ as $\lfloor x \rfloor+\{x\}$ . Expanding everything, we get a quadratic in $\{x\}$ in terms of $\lfloor x \rfloor$ : $\{x\}^2+ (2\lfloor x \rfloor -10,000)\{x\} + \lfloor x \rfloor ^2 = 0$

We use the quadratic formula to solve for $\{x\}$ : $\{x\} = \frac {-2\lfloor x \rfloor + 10,000 \pm \sqrt{ ( 2\lfloor x \rfloor - 10,000 )^2 - 4\lfloor x \rfloor^2 }}{2} = \frac {-2\lfloor x \rfloor + 10,000 \pm \sqrt{ 4\lfloor x \rfloor^2 -40,000 \lfloor x \rfloor + 10,000^2- 4\lfloor x \rfloor^2 }}{2}$

Since $0 \leq \{x\} < 1$ , we get an inequality which we can then solve. After simplifying a lot, we get that $\lfloor x \rfloor^2 + 2\lfloor x \rfloor - 9999 < 0$ .

Solving over the integers, $-101 < \lfloor x \rfloor < 99$ , and since $\lfloor x \rfloor$ is an integer, there are $\boxed{\text{(C)}~199}$ solutions. Each value of $\lfloor x \rfloor$ should correspond to one value of $x$ , so we are done.

Solution 3

Let $x = a+k$ where $a$ is the integer part of $x$ and $k$ is the fractional part of $x$ . We can then rewrite the problem below:

$(a+k)^2 + 10000a = 10000(a+k)$

From here, we get

$(a+k)^2 + 10000a = 10000a + 10000k$

Solving for $a+k = x$

$(a+k)^2 = 10000k$

$x = a+k = \pm100\sqrt{k}$

Because $0 \leq k < 1$ , we know that $a+k$ cannot be less than or equal to $-100$ nor greater than or equal to $100$ . Therefore:

$-99 \leq x \leq 99$

There are $199$ elements in this range, so the answer is $\boxed{\textbf{(C)} \text{ 199}}$ .

Note (not by author): this solution seems to be invalid at first, because one can not determine whether $x$ is an integer or not. However, it actually works because although $x$ itself might not be an integer, it is very close to one, so there are 199 potential $x$ .

Another Note (not by author of previous note): we can actually determine that $x$ =0 is the only possible integer value of $x$ is we set $x$ = $\lfloor x \rfloor$ we end up with $x$ =0 ~YJC64002776

Yet Another Note EXCEPT THIS ONE'S VERY IMPORTANT, AS IT DISPROVES THE PROOF (also not by the author of any previous notes or of the original solution): this solution claims that $-99 \leq x \leq 99$ , which is not entirely true. When $x = 5000-1000\sqrt{26}$ , the equation originally stated in the question holds true. $5000-1000\sqrt{26}$ is roughly $-99.0195$ (rounded to the nearest ten thousandth), which is not in the $-99 \leq x \leq 99$ interval. It is probably by mere coincidence that this solution gives the correct answer, as the solution is not rigorous. Additionally, $x$ is not necessarily integer, so it is not claimable that there are 199 elements in the range of $-99 \leq x \leq 99$ .

Solution 4

Notice the given equation is equivalent to $(\lfloor x \rfloor+\{x\})^2=10,000\{x\}$

Now we know that $\{x\} < 1$ so plugging in $1$ for $\{x\}$ we can find the upper and lower bounds for the values.

$(\lfloor x \rfloor +1)^2 = 10,000(1)$

$(\lfloor x \rfloor +1) = \pm 100$

$\lfloor x \rfloor = 99, -101$

And just like $\textbf{Solution 2}$ , we see that $-101 < \lfloor x \rfloor < 99$ , and since $\lfloor x \rfloor$ is an integer, there are $\boxed{\text{(C)}~199}$ solutions. Each value of $\lfloor x \rfloor$ should correspond to one value of $x$ , so we are done.

Solution 5

Firstly, if $x$ is an integer, then $10,000\lfloor x \rfloor=10,000x$ , so $x$ must be $0$ .

If $0<x<1$ , then we know the following:

$0<x^2<1$

$10,000\lfloor x \rfloor =0$

$0<10,000x<10,000$

Therefore, $0<x^2+10,000\lfloor x \rfloor <1$ , which overlaps with $0<10,000x<10,000$ . This means that there is at least one real solution between $0$ and $1$ . Since $x^2+10,000\lfloor x \rfloor$ increases quadratically and $10,000x$ increases linearly, there is only one solution for this case.

Similarly, if $1<x<2$ , then we know the following:

$1<x^2<4$

$10,000\lfloor x \rfloor =10,000$

$<10,000<10,000x<20,000$

By following similar logic, we can find that there is one solution between $1$ ad $2$ .

We can also follow the same process to find that there are negative solutions for $x$ as well.

There are not an infinite amount of solutions, so at one point there will be no solutions when $n<x<n+1$ for some integer $n$ . For there to be no solutions in a given range means that the range of $10,000\lfloor x \rfloor + x^2$ does not intersect the range of $10,000x$ . $x^2$ will always be positive, and $10,000\lfloor x \rfloor$ is less than $10,000$ less than $10,000x$ , so when $x^2 >= 10,000$ , the equation will have no solutions. This means that there are $99$ positive solutions, $99$ negative solutions, and $0$ for a total of $\boxed{\text{(C)}~199}$ solutions.

~Owen1204

Solution 6 (General Equation)

General solution to this type of equation $f(x, \lfloor x \rfloor) = 0$ :

1. solve  for  to get 
2. apply , solve  to get the domain of 
3. get  from the domain of  because  is integer, then get  from  by 
Note: function  maps  to its floor. By solving , we get function , mapping 's floor to

$x^2 - 10000x + 10000 \lfloor x \rfloor =0$

$x=5000 \pm 100 \sqrt{2500- \lfloor x \rfloor}$ , $\lfloor x \rfloor \le 2500$

$\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$

If $x= 5000 + 100 \sqrt{2500 - \lfloor x \rfloor}$ , $x \ge 5000$ , it contradicts $x < \lfloor x \rfloor + 1 \le 2501$

So $x= 5000 - 100 \sqrt{2500 - \lfloor x \rfloor}$

Let $k = \lfloor x \rfloor$ , $x= 5000 - 100 \sqrt{2500 - k}$

$k \le 5000 - 100 \sqrt{2500 - k} < k + 1$

$0 \le 5000 - k - 100 \sqrt{2500 - k} < 1$

$0 \le 2500 - k - 100 \sqrt{2500 - k} + 2500 < 1$

$0 \le (\sqrt{2500 - k} - 50)^2 < 1$

$-1 < \sqrt{2500 - k} - 50 < 1$

$49 < \sqrt{2500 - k} < 51$

$-101 < k < 99$

So the number of $k$ 's values is $99-(-101)-1=199$ . Because $x=5000-100\sqrt{2500-k}$ , for each value of $k$ , there is a value for $x$ . The answer is $\boxed{\textbf{(C)} 199}$

~isabelchen

Solution 7

Subtracting $10000\lfloor x\rfloor$ from both sides gives $x^2=10000(x-\lfloor x\rfloor)=10000\{x\}$ . Dividing both sides by $10000$ gives $\left(\frac{x}{100}\right)^2=\{x\}<1$ . $\left(\frac{x}{100}\right)^2<1$ when $-100<x<100$ so the answer is $\boxed{199}$ .

~randomdude10807

Solution 8 (Also Gives General Formula For Values of x)

The question wants to know how many values of $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ . This equation can be simplified as follows

$x^2 + 10,000\lfloor x \rfloor = 10,000x$

$x^2 = 10,000(x-\lfloor x \rfloor)$

Notice that $x-\lfloor x \rfloor$ must be greater than or equal to zero, but less than one. Because of that, the right-hand side of the equation must be less than $10000$ . Therefore, $-100<x<100$ .

$x$ can be expressed as $a+b$ , where $a$ is an integer and $b$ is a real number such that $0 \leq b < 1$ . Notice that in order to satisfy the conditions set down for $x$ , $-100 \leq a \leq 99$ .

Substituting $a$ and $b$ into $x^2 = 10000(x-\lfloor x \rfloor)$ , we get

$(a+b)^2 = 10,000b$

$a^2+2ab+b^2 = 10,000b$

Let's try to turn this into a quadratic equation where we're trying to solve for $b$ . Simplifying, we find

$b^2+(2a-10,000)b+a^2 = 0$

Now the value of $b$ depends on the value of $a$ . Our task is to figure out how many values of $a$ will give me valid values of $b$ (a.k.a the values of $a$ that give me $b$ such that $0 \leq b < 1$ ), as that will be our answer. We must also keep in mind that $-100 \leq a \leq 99$ .

Using the quadratic formula, we can find

$\frac{(-2a+10,000) \pm \sqrt{(2a-10,000)^2-4a^2}}{2}$

The equation above represents the values of $b$ in terms of $a$ . Since it represents $b$ , we want values of $a$ such that the equation is between zero and one, zero inclusive. We can then conceive the inequality below.

$0 \leq \frac{(-2a+10,000) \pm \sqrt{(2a-10,000)^2-4a^2}}{2} < 1$

Wow, that looks like a mess! How are we supposed to easily use such a complicated expression? Maybe it would help to simplify it further. First, let's notice a few things.

I want the numerator to be between 0 and 2, 0 inclusive. Of the $\pm$ in the quadratic formula, only the " $-$ " (minus) will work. Why? Because $-2a+10000$ will always be positive (if I plug in the largest possible value of $a$ , $99$ , it will give $9802$ ), and if I want $b$ to be real, which I do, then $\sqrt{(2a-10,000)^2-4a^2}$ should also be positive. $-2a+10000$ will already be much bigger than 2, and adding more on will only make it bigger. It will never be less than 2. It will never work.

I can simplify the numbers under the square root using Difference of Squares. I can turn $\sqrt{(2a-10,000)^2-4a^2}$ into $\sqrt{(2a-10,000)^2-(2a)^2}$ . Applying Difference of Squares, it will eventually simplify out to be $200\sqrt{2500-a}$ .

Cool! Now we've simplified the square root and figured out that it must be a minus sign, not a plus sign. Our new inequality looks like this:

$0 \leq \frac{(-2a+10,000) - 200\sqrt{2500-a}}{2} < 1$

Simplifying the unsimplified fraction farther, we can get this:

$0 \leq -a + 5,000 - 100\sqrt{2500-a} < 1$

Yay! A simpler equation. As long as the value $a$ fits this inequality, the restrictions set on $b$ are satisfied. But wait. What about the restrictions on $a$ ?

By testing out $-100$ and $99$ (and finding that $-100$ gives a value very close to $1$ , and that $99$ gives exactly $1$ ), we can conclude that any value of $-100 \leq$ a $\leq 99$ will satisfy the inequality above as long as it does not make $-a + 5,000 - 100\sqrt{2500-a}$ equal to $1$ . By testing $-100$ and $99$ , you will also find that $a = 99$ will not satisfy the inequality.

This means that the inequality describing all possible values of $a$ is $-100 \leq a < 99$ .

Remember how these values of $a$ give valid values of $b$ that satisfy the desired inequality $0 \leq b < 1$ and how adding up $a$ and $b$ gives $x$ . In summary, remember that the number of values of $a$ is equivalent to the number of values of $x$ .

$a$ must be integer, so the number of values of $a$ that satisfy $-100 \leq a < 99$ is $\boxed{199}$ .

The answer is $\boxed{\textbf{(C)} 199}$ .

~ :)

Note From Author of Solution: you may have noticed that I missed a slight bit by not proving that the thing under the square root in the quadratic formula I made was positive. Don't worry, it wasn't a mistake. It's actually pretty easy to prove. I'll leave it to you to figure out how to do it!

Another Note From Author of Solution: you may also have noticed that I was not very rigorous in proving that the values of $a$ from $-100$ to $98$ inclusive were possible. I plan to come back and fix that, once I figure out how to. If you have any idea, feel free to add your idea!

Video Solution

https://www.youtube.com/watch?v=vHKPbaXwJUE

@@ Line 86: / Line 86: @@
 Another Note (not by author of previous note): we can actually determine that <math>x</math>=0 is the only possible integer value of <math>x</math> is we set <math>x</math>=<math>\lfloor x \rfloor</math> we end up with <math>x</math>=0   ~YJC64002776
-Yet Another Note EXCEPT THIS ONE'S VERY IMPORTANT, AS IT DISPROVES THE PROOF (also not by the author of any previous notes or of the original solution): this solution claims that <math>-99 \leq x \leq 99</math>, which is not entirely true. When x = <math>5000-1000\sqrt{26}</math>, the equation originally stated in the question holds true. <math>5000-1000\sqrt{26}</math> is roughly <math>-99.0195</math> (rounded to the nearest ten thousandth), which is not in the <math>-99 \leq x \leq 99</math> interval. It is probably by mere coincidence that this solution gives the correct answer, as the solution is not rigorous. Additionally, x is not necessarily integer, so it is not claimable that there are 199 elements in the range of <math>-99 \leq x \leq 99</math>.
+Yet Another Note EXCEPT THIS ONE'S VERY IMPORTANT, AS IT DISPROVES THE PROOF (also not by the author of any previous notes or of the original solution): this solution claims that <math>-99 \leq x \leq 99</math>, which is not entirely true. When <math>x = 5000-1000\sqrt{26}</math>, the equation originally stated in the question holds true. <math>5000-1000\sqrt{26}</math> is roughly <math>-99.0195</math> (rounded to the nearest ten thousandth), which is not in the <math>-99 \leq x \leq 99</math> interval. It is probably by mere coincidence that this solution gives the correct answer, as the solution is not rigorous. Additionally, <math>x</math> is not necessarily integer, so it is not claimable that there are 199 elements in the range of <math>-99 \leq x \leq 99</math>.
 ==Solution 4==

2018 AMC 10B (Problems • Answer Key • Resources)
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