2007 iTest Problems/Problem 57

The following problem is from the Ultimate Question of the 2007 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.

Problem

How many positive integers are within $810$ of exactly $\lfloor \sqrt{810} \rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

Solution

This problem is essentially asking for how many $n$ are there $28$ perfect squares from $n-810$ to $n+810$ .

To find the bounds, note that the difference between consecutive perfect squares are odd numbers. As it increases, the distance between perfect squares increase. Let $a$ be the difference between the minimum perfect square and the next perfect square. Since there are $28$ perfect squares in the range, the last difference is $a+54$ , and the sum of the differences is $\tfrac{1}{2} \cdot 28(2a+54)$ . This equals $1620$ , so writing an equation and solving for $a$ yields $a \approx 30$ . The sum of the odd numbers from $1$ to $29$ is $225$ , so we found a place to start.

Using the boundries as reference, we can make a table to find values of $n$ and find the number of perfect squares.

Value of $n$	Value of $n-810$	Value of $n+810$	Smallest PS in bound	Largest PS in bound	Number of PS
$980$	$170$	$1790$	$14^2 = 196$	$42^2 = 1764$	$29$
$1007$	$197$	$1817$	$15^2 = 225$	$42^2 = 1764$	$28$
$1035$	$225$	$1845$	$15^2 = 225$	$42^2 = 1764$	$28$
$1036$	$226$	$1846$	$16^2 = 256$	$42^2 = 1764$	$27$
$1039$	$229$	$1849$	$16^2 = 256$	$43^2 = 1849$	$28$
$1066$	$256$	$1876$	$16^2 = 256$	$43^2 = 1849$	$28$

If $n$ gets below $1007$ , then there will be more perfect squares because $12^2 = 144$ while $41^2 = 1681$ , so more perfect squares would be gained than lost. Similarly, if $n$ gets above $1066$ , then there will be less perfect squares because $18^2 = 324$ while $44^2 = 1936$ , so more perfect squares would be lost than gained.

Based on the table, there are $(1035-1007+1)+(1066-1039+1) = \boxed{57}$ integers that satisfy the criteria.

2007 iTest (Problems, Answer Key)
Preceded by: Problem 56	Followed by: Problem 58
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2007 iTest Problems/Problem 57

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