1997 AMC 12 Problem 13

Attempt Problem 13 of the 1997 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

How many two-digit positive integers

N

have the property that the sum of

N

and the number obtained by reversing the order of the digits of

N

is a perfect square?

4

5

6

7

8

Write

N=10x+y

and add its reversal

Determine when

11(x+y)

, with

1\le x\le9

and

0\le y\le9

, can be square

The sum is

11(x+y).

Since

1\le x+y\le18,

this is square only when

x+y=11,

giving

121.

The tens digit

x

can be any of

2,3,\ldots,9,

with

y=11-x.

There are

8

integers, so E is correct.

Problem 13 in other years