Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 5"
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'''Case 2:''' <math>n</math> has 4 digits and <math>n \ge 3000</math> | '''Case 2:''' <math>n</math> has 4 digits and <math>n \ge 3000</math> | ||
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+ | <math>3000 \le n < 9999</math>, and <math>3^2 \le S(n) \le 3^2+3 \times 9^2</math> | ||
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...ongoing writing of solution... | ...ongoing writing of solution... | ||
~Tomas Diaz. orders@tomasdiaz.com | ~Tomas Diaz. orders@tomasdiaz.com |
Revision as of 15:16, 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
...ongoing writing of solution...
~Tomas Diaz. orders@tomasdiaz.com