Problem 3. 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
## 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