2021 Fall AMC 10B Problems/Problem 19
Problem
Let be the positive integer
, a
-digit number where each digit is a
. Let
be the leading digit of the
th root of
. What is
Solution
For notation purposes, let be the number
with
digits, and let
be the leading digit of
. As an example,
, because
, and the first digit of that is
.
Notice that for all numbers
; this is because
, and dividing by
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 $B(\sqrt[2]{x}} + B(\sqrt[3]{x}} + B(\sqrt[4]{x}} + B(\sqrt[5]{x}} + B(\sqrt[6]{x}}$ (Error compiling LaTeX. Unknown error_msg). From our previous observation, we know that
Therefore,
. We can evaluate
, the leading digit of
, to be
. Therefore,
.
Similarly, we have
Therefore,
. We know
, so
.
Similarly, and
, so
. We also have
and
, so
. Finally,
and
, so
.
We have that .
~ihatemath123
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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