Track / Stage 4 / 95 of 340 #355 of 1964

Problem 355

AMC 12 late, AIME early

Number theory Difficulty 4.7 Find the answer

Example 1. Let $a$ be a digit from 1 to 9. Discuss the number of perfect squares in the sequence $a, \overline{a a}, \bar{a} \overline{a a}, \overline{a a a a}, \cdots$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Analysis: For $a$ equal to $2,3,7,8$ , $a$ equal to $1,5,9$ , and $a$ equal to $4,6$ , respectively. It is not difficult to see that only $1,4,9$ are perfect squares.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.