Here is a math problem from the 1977 USSR Math Olympiad:
Let A be a 2n-digit number. We call this number special if it is a square and a concatenation of two n-digits squares. Also, the first n-digit square can’t start with zero; the second n-digit square can start with zero, but can’t be equal to zero.
- Find all two- and four-digit special numbers.
- Prove that there exists a 20-digit special number.
- Prove that not more than ten 100-digit special numbers exist.
- Prove that there exists a 30-digit special number.
Obviously, these questions are divided into two groups: show the existence and estimate the bound. Furthermore, this problem can be naturally divided into two other groups. Do you see them? The puzzle about special numbers makes a special day today — you get a four-in-one puzzle.Share: