2020 AMC 10A Problems/Problem 16
- The following problem is from both the 2020 AMC 12A #16 and 2020 AMC 10A #16, so both problems redirect to this page.
Problem
A point is chosen at random within the square in the coordinate plane whose vertices are and
. The probability that the point is within
units of a lattice point is
. (A point
is a lattice point if
and
are both integers.) What is
to the nearest tenth
Solution 1
We consider an individual one-by-one block.
If we draw a quarter of a circle from each corner (where the lattice points are located), each with radius , the area covered by the circles should be
. Because of this, and the fact that there are four circles, we write
Solving for , we obtain
, where with
, we get
, and from here, we simplify and see that
~Crypthes
To be more rigorous, note that
since if
then clearly the probability is greater than
. This would make sure the above solution works, as if
there is overlap with the quartercircles.
Solution 2
As in the previous solution, we obtain the equation , which simplifies to
. Since
is slightly more than
,
is slightly less than
. We notice that
is slightly more than
, so
is roughly
~emerald_block
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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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