2021 AMC 10 B Fall

Solution 1:

We can rewrite as . When approximating values, as we will shortly do, the minus one will become negligible so we can ignore it. When we take the power of ten out of the square root, we'll be multiplying by another power of ten, so the leading digit will not change. Thus the leading digit of will be equal to the leading digit Then is the first digit of The final answer is therefore

Solution 2:

For notation purposes, let be the number with 313 digits, and let be the leading digit of . As an example, , because , and the first digit of that is 7 . Notice that for all numbers ; this is because , and dividing by 10 does not affect the leading digit of a number. Similarly, In general, for positive integers and real numbers , it is true that Behind all this complex notation, all that we're really saying is that the first digit of something like has the same first digit as and . The problem asks for From our previous observation, we know that Therefore, . We can evaluate , the leading digit of , to be 2 . Therefore, . Similarly, we have Therefore, . We know , so .

Next, and , so . We also have and , so . Finally, and , so . We have that (A) 8 .