Track / Stage 5 / 215 of 400 #815 of 1964

Problem 815

AIME late

Number theory Difficulty 5.5 Find the answer

Consider the natural number $n=\overline{a b c d}$ and $x=\sqrt{\overline{a b,(c d)}+\overline{b c,(d a)}+\overline{c d,(a b)}+\overline{d a,(b c)}}$ , where $a, b, c, d$ are non-zero digits.

a) Knowing that $x$ is rational, determine the largest possible value that $n$ can take.

b) Show that if $x$ is a natural number, then $n$ has at least two equal digits.

Marius Perianu, Slatina

Your answer

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

Official solution

## Solution and marking scheme

a) $\overline{a b,(c d)}+\overline{b c,(d a)}+\overline{c d,(a b)}+\overline{d a,(b c)}=\frac{100(a+b+c+d)}{9}$

Then $x=\frac{10}{3} \cdot \sqrt{a+b+c+d}$ , so $x$ is rational if and only if $a+b+c+d$ is a perfect square

Among the digits $a, b, c, d$ , at most one can be equal to 9, so $a+b+c+d \in\{4,9,16,25\}$ , and the maximum value of $n$ is obtained for $a+b+c+d=25$ and is 9871

b) If $x$ is a natural number, then $a+b+c+d=9$

Assuming that $a, b, c, d$ are distinct, it would follow that $a+b+c+d \geq 1+2+3+4=10$ , which is a contradiction

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