Difference between revisions of "2023 AMC 10B Problems/Problem 22"
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Technodoggo (talk | contribs) (→Solution (Quick)) |
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| + | == Problem == | ||
| + | |||
| + | How many distinct values of 𝑥 satisfy | ||
| + | <math>\lfloor{x}\rfloor^2-3x=0.</math> | ||
| + | where <math>\lfloor{x}\rfloor</math> denotes the largest integer less than or equal to 𝑥? | ||
| + | |||
== Solution (Quick) == | == Solution (Quick) == | ||
Revision as of 15:52, 15 November 2023
Problem
How many distinct values of 𝑥 satisfy
where 
denotes the largest integer less than or equal to 𝑥?
Solution (Quick)
A quadratic equation can have up to 2 real solutions. With the , it could also help generate another pair. We have to verify that the solutions are real and distinct.
First, we get the trivial solution by ignoring the floor.
, we get 
as our first pair of solutions.
Up to this point, we can rule out A,E.
Next, we see that This implies that 
must be an integer.
We can guess and check 
as 
which yields 
So we got 4 in total 
~Technodoggo
