Two expressions are written on the board:
\[1 + 22 + 333 + 4444 + 55555 + 666666 +7777777 + 88888888 + 999999999\] \[9 + 98 + 987 + 9876 + 98765 + 987654 + 9876543 + 98765432 + 987654321\]
Determine which one is greater or whether the numbers are equal.
Bryn calls the date beautiful if all \(6\) digits of the date entry are different. For example: 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not. How many beautiful dates are there in \(2023\)?
The distance between two villages equals \(999\) kilometres. When you go from one village to the other, every kilometre you see a sign on the road, saying \(0 \mid 999, \, 1\mid 998, \, 2\mid 997, ..., 999\mid 0\). The signs show the distances to the two villages. Find the number of signs that contain only two different digits. For example, the sign \(0\mid999\) contains only two digits, namely \(0\) and \(9\), whereas the sign \(1\mid998\) contains three digits, namely \(1\), \(8\) and \(9\).