AoPS Wiki talk:Problem of the Day/June 15, 2011
Contents
Problem
AoPSWiki:Problem of the Day/June 15, 2011
Solution
Solution 1
We can solve this problem by a bit of trial and error.
We can guess she rode days and we get
since the mean is clearly
and there are
terms.
That's a bit too small.
We can add to
and get
. That's still too small.
Now, we add to get
, the answer we want.
We now count how many numbers are in the following list: .
Adding to the list gives us
.
Dividing by gives us
. Subtracting
gives us
.
Our list has numbers. Since she started on a Monday, we must add
days. Our answer is
Solution 2
On the first day, Jenny rode miles. On the second day, she rode
miles. On the third day, she rode
miles.
This is the sequence which is an arithmetic sequence: first term
, common difference
.
We are trying to find the number of terms such that the
partial sum of the sequence is
.
The formula for the sum of a partial sequence is , where
is the first term,
is the number of terms, and
is the common difference. (Try to derive it!)
Let and
Then we have:
The second root is not an integer, so the workout lasted for days. The
day after Monday is
.