Difference between revisions of "1985 AIME Problems/Problem 10"

Revision as of 22:22, 27 July 2020

Problem

How many of the first 1000 positive integers can be expressed in the form

$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$ ,

where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ ?

Solution

We will be able to reach the same number of integers while $x$ ranges from 0 to 1 as we will when $x$ ranges from $n$ to $n + 1$ for any integer $n$ (Quick proof: $\lfloor 2(n+x)\rfloor + \ldots = \lfloor 2n + 2x\rfloor + \ldots = 2n + \lfloor 2x\rfloor \ldots$ ). Since $\lfloor 2\cdot50 \rfloor + \lfloor 4\cdot50 \rfloor + \lfloor 6\cdot50 \rfloor + \lfloor 8\cdot50 \rfloor = 100 + 200 + 300 + 400$ , the answer must be exactly 50 times the number of integers we will be able to reach as $x$ ranges from 0 to 1, including 1 but excluding 0.

Solution 1

Noting that all of the numbers are even, we can reduce this to any real number $x$ between $0$ to $\frac 12$ , as this will be equivalent to $\frac n2$ to $\frac {n+1}2$ for any integer $n$ (same reasoning as above). So now we only need to test every 10 numbers; and our answer will be 100 times the number of integers we can reach between 1 and 10.

We can now approach this by directly searching for the integers (this solution) or brute forcing all of the cases (next solution):

We can match up the greatest integer functions with one of the partitions of the integer. If we let $x = \frac 12$ then we get the solution $10$ ; now consider when $x < \frac 12$ : $\lfloor 2x \rfloor = 0$ , $\lfloor 4x \rfloor \le 1$ , $\lfloor 6x \rfloor \le 2$ , $\lfloor 8x \rfloor \le 3$ . But according to this the maximum we can get is $1+2+3 = 6$ , so we only need to try the first 6 numbers.

$1$ : Easily possible, for example try plugging in $x =\frac 18$ .
$2$ : Also simple, for example using $\frac 16$ .
$3$ : The partition must either be $1+1+1$ or $1+2$ . If $\lfloor 4x \rfloor = 1$ , then $x \ge \frac 14$ , but then $\lfloor 8x \rfloor \ge 2$ ; not possible; and vice versa to show that the latter partition doesn't work. So we cannot obtain $3$ .
$4$ : We can partition as $1+1+2$ , and from the previous case we see that $\frac 14$ works.
$5$ : We can partition as $1+2+2$ , from which we find that $\frac 13$ works.
$6$ : We can partition as $1+2+3$ , from which we find that $\frac 38$ works.

Out of these 6 cases, only 3 fails. So between 1 and 10 we can reach only the integers $1,2,4,5,6,10$ ; hence our solution is $6 \cdot 100 = \boxed{600}$ .

Solution 2

As we change the value of $x$ , the value of our expression changes only when $x$ crosses rational number of the form $\frac{m}{n}$ , where $n$ is divisible by 2, 4, 6 or 8. Thus, we need only see what happens at the numbers of the form $\frac{m}{\textrm{lcm}(2, 4, 6, 8)} = \frac{m}{24}$ . This gives us 24 calculations to make; we summarize the results here:

$\frac{1}{24}, \frac{2}{24} \to 0$

$\frac{3}{24} \to 1$

$\frac{4}{24}, \frac{5}{24} \to 2$

$\frac{6}{24}, \frac{7}{24} \to 4$

$\frac{8}{24} \to 5$

$\frac{9}{24}, \frac{10}{24}, \frac{11}{24} \to 6$

$\frac{12}{24}, \frac{13}{24}, \frac{14}{24} \to 10$

$\frac{15}{24} \to 11$

$\frac{16}{24},\frac{17}{24} \to 12$

$\frac{18}{24}, \frac{19}{24} \to 14$

$\frac{20}{24}\to 15$

$\frac{21}{24}, \frac{22}{24}, \frac{23}{24} \to16$

$\frac{24}{24} \to 20$

Thus, we hit 12 of the first 20 integers and so we hit $50 \cdot 12 = \boxed{600}$ of the first $1000$ .

Solution 2 Shortcut

Because $2,4,6,8$ are all multiples of $2$ , we can speed things up. We only need to check up to $\frac{12}{24}$ , and the rest should repeat. As shown before, we hit 6 integers ( $1,2,4,5,6,10$ ) from $\frac{1}{24}$ to $\frac{12}{24}$ . Similarly, this should repeat 100 times, for $\boxed{600}$

~N828335

Solution 3

Recall from Hermite's Identity that $\sum_{k = 0}^{n - 1}\left\lfloor x + \frac kn\right\rfloor = \lfloor nx\rfloor$ . Then we can rewrite $\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor = 4\lfloor x\rfloor + \left\lfloor x + \frac18\right\rfloor + \left\lfloor x + \frac16\right\rfloor + 2\left\lfloor x + \frac14\right\rfloor + \left\lfloor x + \frac13\right\rfloor$ $+ \left\lfloor x + \frac38\right\rfloor + 4\left\lfloor x + \frac12\right\rfloor + \left\lfloor x + \frac58\right\rfloor + \left\lfloor x + \frac23\right\rfloor + 2\left\lfloor x + \frac34\right\rfloor + \left\lfloor x + \frac56\right\rfloor + \left\lfloor x + \frac78\right\rfloor$ . There are $12$ terms here (we don't actually have to write all of it out; we can just see where there will be duplicates and subtract accordingly from $20$ ). Starting from every integer $x$ , we can keep adding to achieve one higher value for each of these terms, but after raising the last term, we will have raised the whole sum by $20$ while only achieving $12$ of those $20$ values. We can conveniently shift the $1000$ (since it can be achieved) to the position of the $0$ so that there are only complete cycles of $20$ , and the answer is $\frac {12}{20}\cdot1000 = \boxed{600}$ .

Solution 4

Imagine that we increase $x$ from $0$ to $1$ . At the beginning, the value of our expression is $0$ , at the end it is $2+4+6+8=20$ . How many integers between $1$ and $20$ did we skip? We skip some integers precisely at those points where at least two of $2x$ , $4x$ , $6x$ , and $8x$ become integers at the same time.

Obviously, for $x=1/2$ and $x=1$ all four values become integers at the same time, hence we skip three integers at each of these locations. Additionally, for $x=1/4$ and $x=3/4$ the values $4x$ and $8x$ become integers at the same time, hence we skip one integer at each of the locations.

Therefore for $x\in(0,1]$ we skip a total of $3+3+1+1=8$ integers. As in Solution 2, we conclude that we hit $12$ of the integers from $1$ to $20$ , and so we hit $50 \cdot 12 = \boxed{600}$ of the first $1000$ .

Solution 5

Let $x=\lfloor x\rfloor+\{x\}$ then $\begin{align*} \lfloor 2x\rfloor+\lfloor 4x\rfloor+\lfloor 6x\rfloor+\lfloor 8x\rfloor&=\lfloor 2(\lfloor x\rfloor+\{x\})\rfloor+\lfloor 4(\lfloor x\rfloor+\{x\})\rfloor+\lfloor 6(\lfloor x\rfloor+\{x\})\rfloor+\lfloor 8(\lfloor x\rfloor+\{x\})\rfloor\\ &=2\lfloor x\rfloor+4\lfloor x\rfloor+6\lfloor x\rfloor+8\lfloor x\rfloor+\lfloor 2\{x\}\rfloor+\lfloor 4\{x\}\rfloor+\lfloor 6\{x\}\rfloor+\lfloor 8\{x\}\rfloor\\ &=20\lfloor x\rfloor+(\lfloor 2\{x\}\rfloor+\lfloor 4\{x\}\rfloor+\lfloor 6\{x\}\rfloor+\lfloor 8\{x\}\rfloor) \end{align*}$ Similar to the previous solutions, the value of $\lfloor 2\{x\}\rfloor+\lfloor 4\{x\}\rfloor+\lfloor 6\{x\}\rfloor+\lfloor 8\{x\}\rfloor$ changes when $\{x\}=\frac{m}{n}$ , where $m\in\{1,2,3,...,n-1\}$ , $n\in\{2,4,6,8\}$ . Using Euler's Totient Function $\sum\limits_{k=0}^4 \phi(2k)$ to obtain $12$ different values for $\{x\}=\frac{m}{n}$ . (note that here Euler's Totient Function counts the number of $\{x\}=\frac{m}{n}$ where $m$ , $n$ are relatively prime so that the values of $\{x\}$ won't overlap.).

Thus if $k$ can be expressed as $\lfloor 2x\rfloor+\lfloor 4x\rfloor+\lfloor 6x\rfloor+\lfloor 8x\rfloor$ , then $k=20a+b$ for some non-negative integers $a$ , $b$ , where there are $12$ values for $b$ .

Exclusively, there are $49$ values for $a$ in the range $0<k<1000$ , or $49\cdot12=588$ ordered pairs $(a,b)$ .

If $a=0$ , $b\neq0$ , which includes $11$ ordered pairs.

If $a=50$ , $b=0$ , which includes $1$ ordered pair.

In total, there are $588+11+1=\boxed{600}$ values for $k$ .

~ Nafer

Solution 6

To simplify the question, let $y = 2x$ . Then, the expression in the question becomes $\lfloor y \rfloor + \lfloor 2y \rfloor + \lfloor 3y \rfloor + \lfloor 4y \rfloor$ .

Let $\{x\}$ represent the non-integer part of $x$ (For example, $\{2.8\} = 0.8$ ). Then,

$\begin{align*} \lfloor y \rfloor + \lfloor 2y \rfloor + \lfloor 3y \rfloor + \lfloor 4y \rfloor &= y - \{y\} + 2y - \{2y\} + 3y - \{3y\} + 4y - \{4y\} \\ &= 10y - (\{y\} + \{2y\} + \{3y\} + \{4y\}) \\ &= 10(\lfloor y \rfloor + \{y\}) - (\{y\} + \{2y\} + \{3y\} + \{4y\}) \\ &= 10\lfloor y \rfloor + 10\{y\} - (\{y\} + \{2y\} + \{3y\} + \{4y\}) \\ &= 10\lfloor y \rfloor + 9\{y\} - (\{2y\} + \{3y\} + \{4y\}) \\ \end{align*}$

Since $\lfloor y \rfloor$ is always an integer, $10\lfloor y \rfloor$ will be a multiple of 10. Thus, we look for the range of the other part of the expression. We will be able to reach the same numbers when $y$ ranges from $0$ to $1$ , because the curly brackets ( $\{\}$ ) gets rid of any integer part. Let the combined integer part of $\{2y\} + \{3y\} + \{4y\}$ be $k$ (In other words, $k = \left\lfloor \{2y\} + \{3y\} + \{4y\} \right\rfloor$ ). Then,

$\begin{align*} 9\{y\} - (\{2y\} + \{3y\} + \{4y\}) &= 9\{y\} - (2\{y\} + 3\{y\} + 4\{y\} - k) \\ &= 9\{y\} - (9\{y\} - k) \\ &= k \end{align*}$

The maximum value of $k$ will be when $y$ is slightly less than $1$ , which means $k = 1 + 2 + 3 = 6$ . As $y$ increases from $0$ to $1$ , $k$ will increase whenever $2y$ , $3y$ , or $4y$ is an integer, which happens when $y$ hits one of the numbers in the set $\left\{\dfrac14, \dfrac13, \dfrac12, \dfrac23, \dfrac34 \right\}$ . When $y$ reaches $\dfrac12$ , both $2y$ and $4y$ will be an integer, so $k$ will increase by $2$ . For all the other numbers in the set, $k$ increases by $1$ since only $1$ number in the set will be an integer. Thus, the possible values of $k$ are $\{0, 1, 2, 4, 5, 6\}$ .

Finally, looking back at the original expression, we plug in $k$ to get a multiple of $10$ plus any number in $\{0, 1, 2, 4, 5, 6\}$ . Thus, we hit all numbers ending in $\{0, 1, 2, 4, 5, 6\}$ , of which there are $\boxed{600}$ .

~Owen1204

@@ Line 125: / Line 125: @@
 The maximum value of <math>k</math> will be when <math>y</math> is slightly less than <math>1</math>, which means <math>k = 1 + 2 + 3 = 6</math>. As <math>y</math> increases from <math>0</math> to <math>1</math>, <math>k</math> will increase whenever <math>2y</math>, <math>3y</math>, or <math>4y</math> is an integer, which happens when <math>y</math> hits one of the numbers in the set <math>\left\{\dfrac14, \dfrac13, \dfrac12, \dfrac23, \dfrac34 \right\}</math>. When <math>y</math> reaches <math>\dfrac12</math>, both <math>2y</math> and <math>4y</math> will be an integer, so <math>k</math> will increase by <math>2</math>. For all the other numbers in the set, <math>k</math> increases by <math>1</math> since only <math>1</math> number in the set will be an integer. Thus, the possible values of <math>k</math> are <math>\{0, 1, 2, 4, 5, 6\}</math>.
-Finally, looking back at the original expression, we plug in <math>k</math> to get a multiple of <math>10</math> plus any number in <math>\{0, 1, 2, 4, 5, 6\}</math>. Thus, in each interval between multiples of <math>10</math>, we hit <math>6</math> of the numbers. Since there are <math>100</math> such intervals between <math>0</math> and <math>1000</math>, we can hit <math>\boxed{600}</math> numbers.
+Finally, looking back at the original expression, we plug in <math>k</math> to get a multiple of <math>10</math> plus any number in <math>\{0, 1, 2, 4, 5, 6\}</math>. Thus, we hit all numbers ending in <math>\{0, 1, 2, 4, 5, 6\}</math>, of which there are <math>\boxed{600}</math>.
 ~Owen1204

1985 AIME (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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