2011 AIME II Problems/Problem 15
Contents
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 1
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 cases:
Case :
must be less than the first perfect square after
, which is
, i.e.:
(because
implies
)
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, :
Thus, the answer is .
P.S. You don't need to calculate all the values of P(x) calculated by the above solution. Some very simple modular arithmetic eliminates a large portion of the numbers. The time saved is not that much if you are already at your mathcounts prime.
Solution 2
Make the substitution , so
We're looking for solutions to
with the new bounds
. Since the right side is an integer, it must be that
is a perfect square. For simplicity, write
and
Since
, it must be that
, which gives solutions
, respectively. But this gives us three cases to check:
Case 1: .
In this case, we have
Case 2:
.
In this case, we have
Case 3:
In this case, we have
To finish, the total length of the interval from which we choose
is
. The total length of the success intervals is
which means the probability is
AIMEifying the answer gives
.
See also
2011 AIME II (Problems • Answer Key • Resources) | ||
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