2017 AMC 12A Problems/Problem 9
Problem
Let be the set of points
in the coordinate plane such that two of the three quantities
,
, and
are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of
?
Solution
If the two equal values are and
, then
. Also,
because
is the common value. Solving for
, we get
. Therefore the portion of the line
where
is part of
. This is a ray with an endpoint of
.
Similar to the process above, we assume that the two equal values are and
. Solving the equation
then
. Also,
because
is the common value. Solving for
, we get
. Therefore the portion of the line
where
is also part of
. This is another ray with the same endpoint as the above ray:
.
If and
are the two equal values, then
. Solving the equation for
, we get
. Also
because
is one way to express the common value (using
as the common value works as well). Solving for
, we get
. Therefore the portion of the line
where
is part of
like the other two rays. The lowest possible value that can be achieved is also
.
Since is made up of three rays with common endpoint
, the answer is
.
Solution by TheMathematicsTiger7
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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 |
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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.