Difference between revisions of "2007 AIME II Problems/Problem 7"

Latest revision as of 18:04, 4 December 2020

Problem

Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$

Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

Solution

Solution 1

For $x = 1$ , we see that $\sqrt[3]{1} \ldots \sqrt[3]{7}$ all work, giving 7 integers. For $x=2$ , we see that in $\sqrt[3]{8} \ldots \sqrt[3]{26}$ , all of the even numbers work, giving 10 integers. For $x = 3$ , we get 13, and so on. We can predict that at $x = 22$ we get 70.

To prove this, note that all of the numbers from $\sqrt[3]{x^3} \ldots \sqrt[3]{(x+1)^3 - 1}$ divisible by $x$ work. Thus, $\frac{(x+1)^3 - 1 - x^3}{x} + 1 = \frac{3x^2 + 3x + 1 - 1}{x} + 1 = 3x + 4$ (the one to be inclusive) integers will fit the conditions. $3k + 4 = 70 \Longrightarrow k = 22$ .

The maximum value of $n_i = (x + 1)^3 - 1$ . Therefore, the solution is $\frac{23^3 - 1}{22} = \frac{(23 - 1)(23^2 + 23 + 1)}{22} = 529 + 23 + 1 = 553$ .

Solution 2

Obviously $k$ is positive. Then, we can let $n_1$ equal $k^3$ and similarly let $n_i$ equal $k^3 + (i - 1)k$ .

The wording of this problem (which uses "exactly") tells us that $k^3+69k<(k+1)^3 = k^3 + 3k^2 + 3k+1 \leq k^3+70k$ . Taking away $k^3$ from our inequality results in $69k<3k^2+3k+1\leq 70k$ . Since $69k$ , $3k^2+3k+1$ , and $70k$ are all integers, this inequality is equivalent to $69k\leq 3k^2+3k<70k$ . Since $k$ is positive, we can divide the inequality by $k$ to get $69 \leq 3k+3 < 70$ . Clearly the only $k$ that satisfies is $k=22$ .

Then, $\frac{n_{70}}{k}=k^2+69=484+69=\boxed{553}$ is the maximum value of $\frac{n_i}{k}$ . (Remember we set $n_i$ equal to $k^3 + (i - 1)k$ !)

Video Solution by the Beauty of Math

https://youtu.be/42kXJgD_b-A

@@ Line 1: / Line 1: @@
 == Problem ==
-Given a real number <math>x,</math> let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x.</math> For a certain integer <math>k,</math> there are exactly <math>70</math> positive integers <math>n_{1}, n_{2}, \ldots, n_{70}</math> such that <math>k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{1}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math>
+Given a [[real number]] <math>x,</math> let <math>\lfloor x \rfloor</math> denote the [[floor function|greatest integer]] less than or equal to <math>x.</math> For a certain [[integer]] <math>k,</math> there are exactly <math>70</math> positive integers <math>n_{1}, n_{2}, \ldots, n_{70}</math> such that <math>k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor</math> and <math>k</math> divides <math>n_{i}</math> for all <math>i</math> such that <math>1 \leq i \leq 70.</math>
 Find the maximum value of <math>\frac{n_{i}}{k}</math> for <math>1\leq i \leq 70.</math>
-== Solution ==
+==Solution==
-For <math>x = 1</math>, we see that <math>\sqrt[3]{1} \ldots \sqrt[3]{7}</math> all work, giving 7 integers. For <math>x=2</math>, we see that in <math>\sqrt[3]{8} \ldots \sqrt[3]{26}</math>, all of the even numbers work, giving 10 integers. For <math>x = 3</math>, we get 13, and so on. We can predict that at <math>x = 22</math> we get 70.
+=== Solution 1===
+For <math>x = 1</math>, we see that <math>\sqrt[3]{1} \ldots \sqrt[3]{7}</math> all work, giving 7 integers. For <math>x=2</math>, we see that in <math>\sqrt[3]{8} \ldots \sqrt[3]{26}</math>, all of the [[even]] numbers work, giving 10 integers. For <math>x = 3</math>, we get 13, and so on. We can predict that at <math>x = 22</math> we get 70.
 To prove this, note that all of the numbers from <math>\sqrt[3]{x^3} \ldots \sqrt[3]{(x+1)^3 - 1}</math> divisible by <math>x</math> work. Thus, <math>\frac{(x+1)^3 - 1 - x^3}{x} + 1  = \frac{3x^2 + 3x + 1 - 1}{x} + 1 = 3x + 4</math> (the one to be inclusive) integers will fit the conditions. <math>3k + 4 = 70 \Longrightarrow k = 22</math>.
-The maximum value of <math>\displaystyle n_i = (x + 1)^3 - 1</math>. Therefore, the solution is <math>\frac{23^3 - 1}{22} = \frac{(23 - 1)(23^2 + 23 + 1)}{22} = 529 + 23 + 1 = 553</math>.
+The maximum value of <math>n_i = (x + 1)^3 - 1</math>. Therefore, the solution is <math>\frac{23^3 - 1}{22} = \frac{(23 - 1)(23^2 + 23 + 1)}{22} = 529 + 23 + 1 = 553</math>.
+===Solution 2===
+Obviously <math>k</math> is positive. Then, we can let <math>n_1</math> equal <math>k^3</math> and similarly let <math>n_i</math> equal <math>k^3 + (i - 1)k</math>.
+The wording of this problem (which uses "exactly") tells us that <math>k^3+69k<(k+1)^3 = k^3 + 3k^2 + 3k+1 \leq k^3+70k</math>. Taking away <math>k^3</math> from our inequality results in <math>69k<3k^2+3k+1\leq 70k</math>. Since <math>69k</math>, <math>3k^2+3k+1</math>, and <math>70k</math> are all integers, this inequality is equivalent to <math>69k\leq 3k^2+3k<70k</math>. Since <math>k</math> is positive, we can divide the inequality by <math>k</math> to get <math>69 \leq 3k+3 < 70</math>. Clearly the only <math>k</math> that satisfies is <math>k=22</math>.
+Then, <math>\frac{n_{70}}{k}=k^2+69=484+69=\boxed{553}</math> is the maximum value of <math>\frac{n_i}{k}</math>. (Remember we set <math>n_i</math> equal to <math>k^3 + (i - 1)k</math>!)
+==Video Solution by the Beauty of Math==
+https://youtu.be/42kXJgD_b-A
 == See also ==
@@ Line 15: / Line 26: @@
 [[Category:Intermediate Algebra Problems]]
+{{MAA Notice}}

2007 AIME II (Problems • Answer Key • Resources)
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