SOLUTION: A sequence of digits is constructed by writing the digits of consecutive positive integers (123456789101112 ..... ). What is the digit in the 500th position?

First 9 positions are occupied by one-digit numbers from 1 to 9.


Next 2*90 = 180 positions are occupied by two-digit numbers from 10 to 99.


So, the position from 1 to 189 are just occupied.


Next, we have  500 - 189 = 311 posirions starting from 190th to 500 inclusive.


They are occupied by three-digit numbers, starting from 100.


The number of such 3-digit numbers in positions from 190 to 500 is


    [  ] = 103  (the integer part of  103 ).


So, the positions from  190th  to  189 + 3*103 = 498  are occupied by  103  numbers  from  100  to  202 inclusive.


In positions 499, 500 and 501, we have the number 203; so, the position 500 is occupied by 0 (zer0) digit.     ANSWER