[ [Decimal numeral system ]

Find the largest six-digit number, in which each digit, starting from the third, is equal to the sum of the two preceding digits.

#
Notice that when the number of digits in two numbers is the same, the larger one is the one with the larger first digit.

## Solution

If the first letter was $a$, and the second $-b$, then the third will be $(a+b)$, the fourth $-(a+2b)$, the fifth $-(2a+3b)$, and the sixth $-(3a+5b)$. We need to choose the maximum possible value of $a$ so that the sixth digit remains a "digit," i.e., the inequality $3a+5b<10$ is satisfied. This is possible with $a=3, b=0$, so the desired number will be 303369.

## Answer

303369.