Question 162
We say that a positive integer with an even number of digits is "correct" if, reading its individual digits aloud, we obtain a correct description of the number itself. More precisely, if every digit in an odd position indicates how many times the following digit appears in the whole number.
For example, $1210$ is correct, because it has "one $2$, one $0$", and so is $2121$, because it has "two $1$s, two $1$s", while $1031$ is not, because it claims to have "one $0$, three $1$s", when in fact it has two $1$s.
a. Prove that there are more than $2019$ correct numbers.
b. Prove that the correct integers are finitely many.
c. Find the number of digits of the largest correct number.
For example, $1210$ is correct, because it has "one $2$, one $0$", and so is $2121$, because it has "two $1$s, two $1$s", while $1031$ is not, because it claims to have "one $0$, three $1$s", when in fact it has two $1$s.
a. Prove that there are more than $2019$ correct numbers.
b. Prove that the correct integers are finitely many.
c. Find the number of digits of the largest correct number.