Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 5"
Line 37: | Line 37: | ||
Let <math>k</math> be the 2nd digit of <math>n</math> | Let <math>k</math> be the 2nd digit of <math>n</math> | ||
− | 2000+100k \le n \le 2099+9k, and <math>2^2+k^2 \le S(n) \le 2^2+k^2+2 \times 9^2</math> | + | <math>2000+100k \le n \le 2099+9k</math>, and <math>2^2+k^2 \le S(n) \le 2^2+k^2+2 \times 9^2</math> |
<math>(k-1)100+93 \le n-2007 \le (k-1)100+92</math>, and <math>4+k^2 \le S(n) \le 166+k^2</math> | <math>(k-1)100+93 \le n-2007 \le (k-1)100+92</math>, and <math>4+k^2 \le S(n) \le 166+k^2</math> |
Revision as of 15:37, 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
...ongoing writing of solution...
~Tomas Diaz. orders@tomasdiaz.com