Difference between revisions of "2011 AIME II Problems/Problem 15"
Baijiangchen (talk | contribs) (→Solution) |
Baijiangchen (talk | contribs) m (→Solution) |
||
Line 28: | Line 28: | ||
<math>P(x)</math> must be less than the first perfect square after <math>1</math>, which is <math>4</math>, ''i.e.'': | <math>P(x)</math> must be less than the first perfect square after <math>1</math>, which is <math>4</math>, ''i.e.'': | ||
− | <math>1 \le P(x) < 4</math> | + | <math>1 \le P(x) < 4</math> (because <math>1 \le \sqrt{P(x)} < 2</math>) |
Since <math>P(x)</math> is increasing for <math>x \ge 5</math>, we just need to find the value <math>v \ge 5</math> where <math>P(v) = 4</math>, which will give us the working range <math>5 \le x < v</math>. | Since <math>P(x)</math> is increasing for <math>x \ge 5</math>, we just need to find the value <math>v \ge 5</math> where <math>P(v) = 4</math>, which will give us the working range <math>5 \le x < v</math>. |
Revision as of 18:06, 26 August 2012
Problem
Let . A real number is chosen at random from the interval . The probability that is equal to , where , , , , and are positive integers. Find .
Solution
Table of values of :
$
In order for to hold, must be an integer and hence must be a perfect square. This limits to or or since, from the table above, those are the only values of for which is an perfect square. However, in order for to be rounded down to , must be less than the next perfect square after (for the said intervals). Now, we consider the three difference cases.
Case :
must be less than the first perfect square after , which is , i.e.:
(because )
Since is increasing for , we just need to find the value where , which will give us the working range .
$
So in this case, the only values that will work are .
Case :
must be less than the first perfect square after , which is .
$
So in this case, the only values that will work are .
Case :
must be less than the first perfect square after , which is .
$
So in this case, the only values that will work are .
Now, we find the length of the working intervals and divide it by the length of the total interval, :
$
So the answer is .