Mock AIME 6 2006-2007 Problems/Problem 5

Problem

Let $S(n)$ be the sum of the squares of the digits of $n$ . How many positive integers $n>2007$ satisfy the inequality $n-S(n)\le 2007$ ?

We start by rearranging the inequality the following way:

$n-2007\le S(n)$ and compare the possible values for the left hand side and the right hand side of this innequality.

Case 1: $n$ has 5 digits or more.

Let $d$ = number of digits of n.

Then as a function of d,

$10^d \le n < 10^{d+1}-1$ , and $1 \le S(n) \le 9^2d$

$10^d - 2007 \le n < 10^{d+1}-2008$ , and $1 \le S(n) \le 81d$

when $d \ge 5$ ,

$10^d - 2007 \ge 10^5 -2007$

$10^d - 2007 \ge 10^5 -2007 > 81d$

Since $10^d - 2007 > 81d$ for $d \ge 5$ , then $n-2007\not\le S(n)$ and there is no $S(n)$ possible when $n$ has 5 or more digits.

~Tomas Diaz. orders@tomasdiaz.com