2001 AIME II Problems/Problem 1

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Problem

Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$ ?

Solution

The two-digit perfect squares are $16, 25, 36, 49, 64, 81$ . We try making a sequence starting with each one:

$16 - 64 - 49$ . This terminates since none of them end in a $9$ , giving us $1649$ .
$25$ .
$36 - 64 - 49$ , $3649$ .
$49$ .
$64 - 49$ , $649$ .
$81 - 16 - 64 - 49$ , $81649$ .

The largest is $81649$ , so our answer is $\boxed{816}$ .

2001 AIME II (Problems • Resources)
Preceded by First question	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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2001 AIME II Problems/Problem 1

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