Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 5"
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Substituting for all values of a in the above inequality we get: | Substituting for all values of a in the above inequality we get: | ||
− | When <math>a=0,\;</math>, <math>0 \le b^2-b+11</math>, which gives: <math>0 \le b \le 9</math>. | + | When <math>a=0,\;</math>, <math>0 \le b^2-b+11</math>, which gives: <math>0 \le b \le 9</math>. But <math>n>2007&, So, </math>n=2008<math> and </math>n=2009<math> Total possible </math>n<math>'s: '''2''' |
− | When <math>a=1,\;< | + | When </math>a=1,\;<math>, </math>0 \le b^2-b+2<math>, which gives: </math>0 \le b \le 9<math>. Total possible </math>n<math>'s: '''10''' |
− | When <math>a=2,\;< | + | When </math>a=2,\;<math>, </math>0 \le b^2-b-5<math>, which gives: </math>3 \le b \le 9<math>. Total possible </math>n<math>'s: '''7''' |
− | When <math>a=3,\;< | + | When </math>a=3,\;<math>, </math>0 \le b^2-b-10<math>, which gives: </math>4 \le b \le 9<math>. Total possible </math>n<math>'s: '''6''' |
− | When <math>a=4,\;< | + | When </math>a=4,\;<math>, </math>0 \le b^2-b-13<math>, which gives: </math>5 \le b \le 9<math>. Total possible </math>n<math>'s: '''5''' |
− | When <math>a=5,\;< | + | When </math>a=5,\;<math>, </math>0 \le b^2-b-14<math>, which gives: </math>5 \le b \le 9<math>. Total possible </math>n<math>'s: '''5''' |
− | When <math>a=6,\;< | + | When </math>a=6,\;<math>, </math>0 \le b^2-b-13<math>, which gives: </math>5 \le b \le 9<math>. Total possible </math>n<math>'s: '''5''' |
− | When <math>a=7,\;< | + | When </math>a=7,\;<math>, </math>0 \le b^2-b-10<math>, which gives: </math>4 \le b \le 9<math>. Total possible </math>n<math>'s: '''6''' |
− | When <math>a=8,\;< | + | When </math>a=8,\;<math>, </math>0 \le b^2-b-5<math>, which gives: </math>3 \le b \le 9<math>. Total possible </math>n<math>'s: '''7''' |
− | When <math>a=9,\;< | + | When </math>a=9,\;<math>, </math>0 \le b^2-b+2<math>, which gives: </math>0 \le b \le 9<math>. Total possible </math>n<math>'s: '''10''' |
− | Therefore, the total number of possible <math>n< | + | Therefore, the total number of possible </math>n<math>'s is: </math>2+10+7+6+5+5+5+6+7+10=\boxed{63}$ |
~Tomas Diaz. orders@tomasdiaz.com | ~Tomas Diaz. orders@tomasdiaz.com | ||
{{alternate solutions}} | {{alternate solutions}} |
Revision as of 22:56, 24 November 2023
Problem
Let be the sum of the squares of the digits of
. How many positive integers
satisfy the inequality
?
Solution
We start by rearranging the inequality the following way:
and compare the possible values for the left hand side and the right hand side of this inequality.
Case 1: has 5 digits or more.
Let = number of digits of n.
Then as a function of d,
, and
, and
when ,
Since for
, then
and there is no possible
when
has 5 or more digits.
Case 2: has 4 digits and
, and
, and
Since , then
and there is no possible
when
has 4 digits and
.
Case 3:
Let be the 2nd digit of
, and
, and
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
Since , for
, then
and there is no possible
when
when combined with the previous cases.
Case 4:
Let be the 3rd digit of
, and
, and
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
At ,
,
.
Since , for
, then
and there is no possible
when
when combined with the previous cases.
From cases 1 through 4 we now know that
Case 5:
Let and
be the 3rd and 4th digits of n respectively with
and
;
Solving the inequality we have:
Substituting for all values of a in the above inequality we get:
When ,
, which gives:
. But $n>2007&, So,$ (Error compiling LaTeX. Unknown error_msg)n=2008
n=2009
n$'s: '''2'''
When$ (Error compiling LaTeX. Unknown error_msg)a=1,\;0 \le b^2-b+2
0 \le b \le 9
n$'s: '''10'''
When$ (Error compiling LaTeX. Unknown error_msg)a=2,\;0 \le b^2-b-5
3 \le b \le 9
n$'s: '''7'''
When$ (Error compiling LaTeX. Unknown error_msg)a=3,\;0 \le b^2-b-10
4 \le b \le 9
n$'s: '''6'''
When$ (Error compiling LaTeX. Unknown error_msg)a=4,\;0 \le b^2-b-13
5 \le b \le 9
n$'s: '''5'''
When$ (Error compiling LaTeX. Unknown error_msg)a=5,\;0 \le b^2-b-14
5 \le b \le 9
n$'s: '''5'''
When$ (Error compiling LaTeX. Unknown error_msg)a=6,\;0 \le b^2-b-13
5 \le b \le 9
n$'s: '''5'''
When$ (Error compiling LaTeX. Unknown error_msg)a=7,\;0 \le b^2-b-10
4 \le b \le 9
n$'s: '''6'''
When$ (Error compiling LaTeX. Unknown error_msg)a=8,\;0 \le b^2-b-5
3 \le b \le 9
n$'s: '''7'''
When$ (Error compiling LaTeX. Unknown error_msg)a=9,\;0 \le b^2-b+2
0 \le b \le 9
n$'s: '''10'''
Therefore, the total number of possible$ (Error compiling LaTeX. Unknown error_msg)n2+10+7+6+5+5+5+6+7+10=\boxed{63}$
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.