2018 AMC 10B Problems/Problem 25
Problem
Let denote the greatest integer less than or equal to
. How many real numbers
satisfy the equation
?
Solution 1
This rewrites itself to .
Graphing and
we see that the former is a set of line segments with slope
from
to
with a hole at
, then
to
with a hole at
etc.
Here is a graph of and
for visualization.
Now notice that when then graph has a hole at
which the equation
passes through and then continues upwards. Thus our set of possible solutions is bounded by
. We can see that
intersects each of the lines once and there are
lines for an answer of
.
Note: from the graph we can clear see there are 4 solution on the negative side and only 2 on the positive side. So the solution really should be from -100 to 98, which still counts to 199.
Solution 2
Same as the first solution, .
We can write as
. Expanding everything, we get a quadratic in
in terms of
:
We use the quadratic formula to solve for {x}:
Since , we get an inequality which we can then solve. After simplifying a lot, we get that
.
Solving over the integers, , and since
is an integer, there are
solutions. Each value of
should correspond to one value of
, so we are done.
Solution 3
Let where
is the integer part of
and
is the fractional part of
.
We can then rewrite the problem below:
From here, we get
Solving for
Because , we know that
cannot be less than or equal to
nor greater than or equal to
. Therefore:
There are 199 elements in this range, so the answer is .
Solution 4
Notice the given equation is equivilent to
Now we now that so plugging in
for
we can find the upper and lower bounds for the values.
And just like Solution 2, we see that , and since
is an integer, there are
solutions. Each value of
should correspond to one value of
, so we are done.
Solution 5
First, we can let . We know that
by definition. We can rearrange the equation to obtain
.
By taking square root on both sides, we obtain (because
). We know since
is the fractional part of
, it must be that
. Thus,
may take any value in the interval
. Hence, we know that there are
potential values for
in that range and we are done.
~awesome1st
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
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 |
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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 12 Problems and Solutions |
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