Difference between revisions of "2007 AIME II Problems/Problem 7"
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== Problem == | == 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_{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> |
Find the maximum value of <math>\frac{n_{i}}{k}</math> for <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. | + | 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>. | 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>. | ||
Revision as of 19:50, 29 March 2007
Problem
Given a real number 
let 
denote the greatest integer less than or equal to 
For a certain integer 
there are exactly 
positive integers 
such that ![$k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{1}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$](http://latex.artofproblemsolving.com/2/a/5/2a5a9731862eee20d3412a0856ff0736dcce16ad.png)
and 
divides 
for all 
such that 
Find the maximum value of 
for 
Solution
For 
, we see that ![$\sqrt[3]{1} \ldots \sqrt[3]{7}$](http://latex.artofproblemsolving.com/9/3/3/9339e15771e9f6fd053bad1f11137bcb1e576985.png)
all work, giving 7 integers. For 
, we see that in ![$\sqrt[3]{8} \ldots \sqrt[3]{26}$](http://latex.artofproblemsolving.com/0/b/1/0b1b2a0bb4dfe13cdca56c74fc2872c55cc07fc7.png)
, all of the even numbers work, giving 10 integers. For 
, we get 13, and so on. We can predict that at 
we get 70.
To prove this, note that all of the numbers from ![$\sqrt[3]{x^3} \ldots \sqrt[3]{(x+1)^3 - 1}$](http://latex.artofproblemsolving.com/d/d/4/dd4c7646f276f40c0128e3ad6cdc26603f11629c.png)
divisible by 
work. Thus, 
(the one to be inclusive) integers will fit the conditions. 
.
The maximum value of 
. Therefore, the solution is 
.
See also
| 2007 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
