Difference between revisions of "2002 AIME II Problems/Problem 8"

Latest revision as of 13:22, 14 July 2023

Problem

Find the least positive integer $k$ for which the equation $\left\lfloor\frac{2002}{n}\right\rfloor=k$ has no integer solutions for $n$ . (The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$ .)

Solutions

Solution 1

Note that if $\frac{2002}n - \frac{2002}{n+1}\leq 1$ , then either $\left\lfloor\frac{2002}{n}\right\rfloor=\left\lfloor\frac{2002}{n+1}\right\rfloor$ , or $\left\lfloor\frac{2002}{n}\right\rfloor=\left\lfloor\frac{2002}{n+1}\right\rfloor+1$ . Either way, we won't skip any natural numbers.

The greatest $n$ such that $\frac{2002}n - \frac{2002}{n+1} > 1$ is $n=44$ . (The inequality simplifies to $n(n+1)<2002$ , which is easy to solve by trial, as the solution is obviously $\simeq \sqrt{2002}$ .)

We can now compute: $\left\lfloor\frac{2002}{45}\right\rfloor=44$ $\left\lfloor\frac{2002}{44}\right\rfloor=45$ $\left\lfloor\frac{2002}{43}\right\rfloor=46$ $\left\lfloor\frac{2002}{42}\right\rfloor=47$ $\left\lfloor\frac{2002}{41}\right\rfloor=48$ $\left\lfloor\frac{2002}{40}\right\rfloor=50$

From the observation above (and the fact that $\left\lfloor\frac{2002}{2002}\right\rfloor=1$ ) we know that all integers between $1$ and $44$ will be achieved for some values of $n$ . Similarly, for $n<40$ we obviously have $\left\lfloor\frac{2002}{n}\right\rfloor > 50$ .

Hence the least positive integer $k$ for which the equation $\left\lfloor\frac{2002}{n}\right\rfloor=k$ has no integer solutions for $n$ is $\boxed{049}$ .

Note

After getting that $\left\lfloor\frac{2002}{45}\right\rfloor=44$ , for ease of computation above, we can use the fact that $(40+k)(49-k)$ varies solely based on $k^2$ and checking these gives us that the pattern fails at $k=0$ giving us $\boxed{049}$ as the answer.

~Dhillonr25

Solution 2

Rewriting the given information and simplifying it a bit, we have $\begin{align*} k \le \frac{2002}{n} < k+1 &\implies \frac{1}{k} \ge \frac{n}{2002} > \frac{1}{k+1}. \\ &\implies \frac{2002}{k} \ge n > \frac{2002}{k+1}. \end{align*}$

Now note that in order for there to be no integer solutions to $n,$ we must have $\left\lfloor \frac{2002}{k} \right\rfloor = \left\lfloor \frac{2002}{k+1} \right\rfloor.$ We seek the smallest such $k.$ A bit of experimentation yields that $k=49$ is the smallest solution, as for $k=49,$ it is true that $\left\lfloor \frac{2002}{49} \right\rfloor = \left\lfloor \frac{2002}{50} \right\rfloor = 40.$ Furthermore, $k=49$ is the smallest such case. (If unsure, we could check if the result holds for $k=48,$ and as it turns out, it doesn't.) Therefore, the answer is $\boxed{049}.$

Solution 3

In this solution we use inductive reasoning and a lot of trial and error. Depending on how accurately you can estimate, the solution will come quicker or slower.

Using values of $k$ as $1, 2, 3, 4,$ and $5,$ we can find the corresponding values of $n$ relatively easily. For $k = 1$ , $n$ is in the range $[2002-1002]$ ; for $k = 2$ , $n$ is the the range $[1001-668]$ , etc: $3, [667,501]; 4, [500-401]; 5, [400-334]$ . For any positive integer $k, n$ is in a range of $\left\lfloor \frac{2002}{k} \right\rfloor -\left\lceil \frac{2002}{k+1} \right\rceil$ .

Now we try testing $k = 1002$ to get a better understanding of what our solution will look like. Obviously, there will be no solution for $n$ , but we are more interested in how the range will compute to. Using the formula we got above, the range will be $1-2$ . Testing any integer $k$ from $1002-2000$ will result in the same range. Also, notice that each and every one of them have no solution for $n$ . Testing $1001$ gives a range of $2-2$ , and $2002$ gives $1-1$ . They each have a solution for $n$ , and their range is only one value. Therefore, we can assume with relative safety that the integer $k$ we want is the lowest integer that follows this equation

$\left\lfloor\frac{2002}{k}\right\rfloor + 1 = \left\lceil \frac{2002}{k+1}\right\rceil$

Now we can easily guess and check starting from $k = 1$ . After a few tests it's not difficult to estimate a few jumps, and it took me only a few minutes to realize the answer was somewhere in the forties (You could also use the fact that $45^2=2025$ ). Then it's just a matter of checking them until we get $\boxed{049}$ . Alternatively, you could use the equation above and proceed with one of the other two solutions listed.

-jackshi2006

Edited and $\LaTeX$ ed by PhunsukhWangdu

Solution 4

Here is an intuitive way to approximate the answer is around $45$ : For the function $f(n)=\frac{2002}{n}$ its derivative is $-\frac{2002}{n^2}$ , which should be close to $-1$ because we need to find the smallest skipped integer. The rest of the steps are the same as Solution 1.

-maxamc

@@ Line 21: / Line 21: @@
 Hence the least positive integer <math>k</math> for which the equation <math>\left\lfloor\frac{2002}{n}\right\rfloor=k</math> has no integer solutions for <math>n</math> is <math>\boxed{049}</math>.
+====Note====
+After getting that <math>\left\lfloor\frac{2002}{45}\right\rfloor=44</math>, for ease of computation above, we can use the fact that <math>(40+k)(49-k)</math> varies solely based on <math>k^2</math> and checking these gives us that the pattern fails at <math>k=0</math> giving us <math>\boxed{049}</math> as the answer.
+~Dhillonr25
 ===Solution 2===
@@ Line 33: / Line 39: @@
 In this solution we use inductive reasoning and a lot of trial and error. Depending on how accurately you can estimate, the solution will come quicker or slower.
-Using values of k as 1, 2, 3, 4, and 5, we can find the corresponding values of n relatively easily. For k = 1, n is in the range [2002-1002]; for k = 2, n is the the range [1001-668], etc: 3, [667,501]; 4, [500-401]; 5, [400-334]. For any positive integer k, n is in a range of <math>floor[2002/k]-ceiling[2002/(k+1)]</math>.
+Using values of <math>k</math> as <math>1, 2, 3, 4,</math> and <math>5,</math> we can find the corresponding values of <math>n</math> relatively easily. For <math>k = 1</math>, <math>n</math> is in the range <math>[2002-1002]</math>; for <math>k = 2</math>, <math>n</math> is the the range <math>[1001-668]</math>, etc: <math>3, [667,501]; 4, [500-401]; 5, [400-334]</math>. For any positive integer <math>k, n</math> is in a range of <math>\left\lfloor \frac{2002}{k} \right\rfloor -\left\lceil \frac{2002}{k+1} \right\rceil</math>.
-Now we try testing k = 1002 to get a better understanding of what our solution will look like. Obviously, there will be no solution for n, but we are more interested in how the range will compute to. Using the formula we got above, the range will be 1-2. Testing any integer k from 1002-2000 will result in the same range. Also, notice that each and every one of them have no solution for n. Testing 1001 gives a range of 2-2, and 2002 gives 1-1. They each have a solution for n, and their range is only one value. Therefore, we can assume with relative safety that the integer k we want is the lowest integer that follows this equation:
+Now we try testing <math>k = 1002</math> to get a better understanding of what our solution will look like. Obviously, there will be no solution for <math>n</math>, but we are more interested in how the range will compute to. Using the formula we got above, the range will be <math>1-2</math>. Testing any integer <math>k</math> from <math>1002-2000</math> will result in the same range. Also, notice that each and every one of them have no solution for <math>n</math>. Testing <math>1001</math> gives a range of <math>2-2</math>, and <math>2002</math> gives <math>1-1</math>. They each have a solution for <math>n</math>, and their range is only one value. Therefore, we can assume with relative safety that the integer <math>k</math> we want is the lowest integer that follows this equation
-floor[2002/k] + 1 = ceiling[2002/(k+1)]
+<cmath>\left\lfloor\frac{2002}{k}\right\rfloor + 1 = \left\lceil \frac{2002}{k+1}\right\rceil</cmath>
-Now we can easily guess and check starting from k = 1. After a few tests it's not difficult to estimate a few jumps, and it took me only a few minutes to realize the answer was somewhere in the forties. Then it's just a matter of checking them until we get <math>\boxed{049}</math>.
+Now we can easily guess and check starting from <math>k = 1</math>. After a few tests it's not difficult to estimate a few jumps, and it took me only a few minutes to realize the answer was somewhere in the forties (You could also use the fact that <math>45^2=2025</math>). Then it's just a matter of checking them until we get <math>\boxed{049}</math>.
 Alternatively, you could use the equation above and proceed with one of the other two solutions listed.
 -jackshi2006
+Edited and <math>\LaTeX</math>ed by PhunsukhWangdu
+===Solution 4===
+Here is an intuitive way to approximate the answer is around <math>45</math>: For the function
+<math>f(n)=\frac{2002}{n}</math>
+its derivative is
+<math>-\frac{2002}{n^2}</math>,
+which should be close to <math>-1</math> because we need to find the smallest skipped integer. The rest of the steps are the same as Solution 1.
+-maxamc
 == See also ==
 {{AIME box|year=2002|n=II|num-b=7|num-a=9}}
 {{MAA Notice}}

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