2007 AIME I Problems/Problem 5
Problem
The formula for converting a Fahrenheit temperature to the corresponding Celsius temperature
is
An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
For how many integer Fahrenheit temperatures between 32 and 1000 inclusive does the original temperature equal the final temperature?
Contents
Solution
Solution 1
Examine modulo 9.
- If
, then we can define
. This shows that
. This case works.
- If
, then we can define
. This shows that
. So this case doesn't work.
Generalizing this, we define that . Thus,
. We need to find all values
that
. Testing every value of
shows that
, so
of every
values of
work.
There are cycles of
, giving
numbers that work. Of the remaining
numbers from
onwards,
work, giving us
as the solution.
Solution 2
Notice that holds if
for some integer
.
Thus, after translating from
we want count how many values of
there are such that
is an integer from
to
. This value is computed as
, adding in the extra solution corresponding to
.
Note
Proof that iff
for some integer
:
First assume that cannot be written in the form
for any integer
. Let
. Our equation simplifies to
. However, this equation is not possible, as we defined
such that it could not be written in this form. Therefore, if
, then
.
Now we will prove that if ,
. We realize that because of the 5 in the denominator of
,
will be at most
away from
. Let
, meaning that
. Now we substitute this into our equation:
.
Now we use the fact that
Hence and we are done.
- mako17
Solution 3
Let be a degree Celsius, and
rounded to the nearest integer. Since
was rounded to the nearest integer we have
, which is equivalent to
if we multiply by
. Therefore, it must round to
because
so
is the closest integer. Therefore there is one solution per degree celcius in the range from
to
, meaning there are
solutions.
Solution 4
Start listing out values for and their corresponding values of
. You will soon find that every 9 values starting from
= 32, there is a pattern:
: Works
: Doesn't work
: work
: Doesn’t work
: Works
: Works
: Doesn’t work
: Works
: Doesn’t work
: Works
There are numbers between
and
, inclusive. This is
sets of
, plus
extra numbers at the end. In each set of
, there are
“Works,” so we have
values of
that work.
Now we must add the extra numbers. The number of “Works” in the first
terms of the pattern is
, so our final answer is
solutions that work.
Submitted by warriorcats
See also
2007 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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