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.
Contents
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
Solutions
Solution 1
Diagram
Diagram by MathandSki Using Asymptote
Note: The diagram represents each unit square of the given square.
Solution
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
Solution 3 (Estimating)
As above, we find that we need to estimate .
Note that we can approximate and so
.
And so our answer is .
~Silverdragon
Solution 4 (Estimating but a bit different)
We only need to figure out the probability for a unit square, as it will scale up to the square. Since we want to find the probability that a point inside a unit square that is
units away from a lattice point (a corner of the square) is
, we can find which answer will come the closest to covering
of the area.
Since the closest is which turns out to be
which is about
, we find that the answer rounded to the nearest tenth is
or
.
~RuiyangWu
Solution 5 (Estimating but differently again)
As per the above diagram, realize that , so
.
.
is between
and
and
, so we can say
.
So . This is slightly above
, since
.
-Solution by Joeya
Video Solution
Video Solution 1
Education, The Study of Everything
Video Solution 2
~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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.