Competition problem — Number theory

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Number theory Difficulty 5.2 AIME, harder Find the answer

1. Below is the inscription by the famous Chinese mathematician, Academician Wang Yuan:
数棈妻好

If different Chinese characters represent different digits from $0 \sim 9$ , and assuming “数学竞赛好” represents the largest five-digit number that is a perfect square, formed by different digits. Then this five-digit number is $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Notice that, $300^{2}=90000,310^{2}=96100$ ,
$320^{2}=102400>100000 \text {. }$

Thus, this five-digit number with distinct digits might be among the nine square numbers between $310^{2} \sim$ $320^{2}$ .
$\begin{array}{l} \text { Also, } 311^{2}=96721,312^{2}=97344,313^{2}=97969, \\ 314^{2}=98596,315^{2}=99225,316^{2}=99856, \\ 317^{2}=100489>100000, \end{array}$

Therefore, the largest square number among the five-digit numbers with distinct digits is 96721.

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