Library / /7 of 11

, 2022

Algebra Difficulty 7.6 National olympiad, round 2 Prove it Hong Kong

A special calculator contains a red button, which counts the number of even digits of an integer. For instance, when the screen shows $2022$ , pressing the red button gives $4$ since all $4$ digits of $2022$ are even. Someone inputs a positive integer $n$ into the calculator and keeps pressing the red button until $0$ is displayed on the screen. If $0$ is displayed after the red button has been pressed four times, find the smallest possible value of $n$ .

Solution

Answer: $2 \times 10^{19}$
Suppose the sequence of numbers shown on the calculator screen is
$n \to p \to q \to r \to 0$
with $p$ , $q$ , $r$ nonzero. Note that $r$ is at least $1$ and so $q$ consists of at least one even digit, which means $q \geq 2$ . Hence $p$ consists of at least two even digits and is no less than $20$ . It follows that $n$ has at least $20$ even digits, and the smallest such number is $2 \times 10^{19}$ , which one easily verifies to be a possible value of $n$ . (The sequence in this case will be $20\ldots00 \to 20 \to 2 \to 1 \to 0$ .)

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.