Difference between revisions of "1985 AIME Problems/Problem 10"
(→Solution: reduce by 1/2) |
(→Solution) |
||
Line 10: | Line 10: | ||
=== Solution 1 === | === Solution 1 === | ||
− | Noting that all of the numbers are even, we can reduce this to any real number <math>x</math> between <math>0</math> to <math>\frac 12</math>, as this will be equivalent to <math> | + | Noting that all of the numbers are even, we can reduce this to any real number <math>x</math> between <math>0</math> to <math>\frac 12</math>, as this will be equivalent to <math>\frac n2</math> to <math>\frac {n+1}2</math> for any integer <math>n</math> (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 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 <math>x = \frac 12</math> then we get the solution <math>10</math>; now consider when <math>x < \frac 12</math>: <math>\lfloor 2x \rfloor = 0</math>, <math>\lfloor 4x \rfloor \le 1</math>, <math>\lfloor 6x \rfloor \le 2</math>, <math> | + | We can match up the greatest integer functions with one of the partitions of the integer. If we let <math>x = \frac 12</math> then we get the solution <math>10</math>; now consider when <math>x < \frac 12</math>: <math>\lfloor 2x \rfloor = 0</math>, <math>\lfloor 4x \rfloor \le 1</math>, <math>\lfloor 6x \rfloor \le 2</math>, <math>\lfloor 8x \rfloor \le 3</math>. But according to this the maximum we can get is <math>1+2+3 = 6</math>, so we only need to try the first 6 numbers. |
− | *<math>1</math>: Easily possible, for example try plugging in <math>x = | + | *<math>1</math>: Easily possible, for example try plugging in <math>x =\frac 18</math>. |
− | *<math>2</math>: Also simple, for example using <math> | + | *<math>2</math>: Also simple, for example using <math>\frac 16</math>. |
*<math>3</math>: The partition must either be <math>1+1+1</math> or <math>1+2</math>. If <math>\lfloor 4x \rfloor = 1</math>, then <math>x \ge \frac 14</math>, but then <math>\lfloor 8x \rfloor \ge 2</math>; not possible; and vice versa to show that the latter partition doesn't work. So we cannot obtain <math>3</math>. | *<math>3</math>: The partition must either be <math>1+1+1</math> or <math>1+2</math>. If <math>\lfloor 4x \rfloor = 1</math>, then <math>x \ge \frac 14</math>, but then <math>\lfloor 8x \rfloor \ge 2</math>; not possible; and vice versa to show that the latter partition doesn't work. So we cannot obtain <math>3</math>. | ||
*<math>4</math>: We can partition as <math>1+1+2</math>, and from the previous case we see that <math>\frac 14</math> works. | *<math>4</math>: We can partition as <math>1+1+2</math>, and from the previous case we see that <math>\frac 14</math> works. | ||
Line 55: | Line 55: | ||
Thus, we hit 12 of the first 20 integers and so we hit <math>50 \cdot 12 = 600</math> of the first 1000. | Thus, we hit 12 of the first 20 integers and so we hit <math>50 \cdot 12 = 600</math> of the first 1000. | ||
+ | |||
+ | ===Solution 3=== | ||
+ | |||
+ | Recall from Hermite's Identity that <math>\sum_{k = 0}^{n - 1}\left\lfloor x + \frac kn\right\rfloor = \lfloor nx\rfloor</math>. Then we can rewrite <math>\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</math> | ||
+ | <math>+ \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</math>. There are <math>12</math> 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 <math>20</math>). Starting from every integer <math>x</math>, 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 <math>20</math> while only achieving <math>12</math> of those <math>20</math> values. We can conveniently shift the <math>1000</math> (since it can be achieved) to the position of the <math>0</math> so that there are only complete cycles of <math>20</math>, and the answer is <math>\frac {12}{20}\cdot1000 = \boxed{600}</math>. | ||
== See also == | == See also == |
Revision as of 22:02, 16 February 2009
Problem
How many of the first 1000 positive integers can be expressed in the form
,
where is a real number, and
denotes the greatest integer less than or equal to
?
Solution
We will be able to reach the same number of integers while ranges from 0 to 1 as we will when
ranges from
to
for any integer
(Quick proof:
). Since
, the answer must be exactly 50 times the number of integers we will be able to reach as
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 between
to
, as this will be equivalent to
to
for any integer
(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 then we get the solution
; now consider when
:
,
,
,
. But according to this the maximum we can get is
, so we only need to try the first 6 numbers.
: Easily possible, for example try plugging in
.
: Also simple, for example using
.
: The partition must either be
or
. If
, then
, but then
; not possible; and vice versa to show that the latter partition doesn't work. So we cannot obtain
.
: We can partition as
, and from the previous case we see that
works.
: We can partition as
, from which we find that
works.
: We can partition as
, from which we find that
works.
Out of these 6 cases, only 3 fails. So between 1 and 10 we can reach only the integers ; hence our solution is
.
Solution 2
As we change the value of , the value of our expression changes only when
crosses rational number of the form
, where
is divisible by 2, 4, 6 or 8. Thus, we need only see what happens at the numbers of the form
. This gives us 24 calculations to make; we summarize the results here:
Thus, we hit 12 of the first 20 integers and so we hit of the first 1000.
Solution 3
Recall from Hermite's Identity that . Then we can rewrite
. There are
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
). Starting from every integer
, 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
while only achieving
of those
values. We can conveniently shift the
(since it can be achieved) to the position of the
so that there are only complete cycles of
, and the answer is
.
See also
1985 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |