Difference between revisions of "1996 AHSME Problems/Problem 27"
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Because both spheres have their centers on the x-axis, we can simplify the graph a bit by looking at a two-dimensional plane (the previous z-axis is the new x-axis and the y-axis remains the same). | Because both spheres have their centers on the x-axis, we can simplify the graph a bit by looking at a two-dimensional plane (the previous z-axis is the new x-axis and the y-axis remains the same). | ||
− | The spheres now become circles with centers at <math>(1,0)</math> and (21/2,0)<math>. They have radii < | + | The spheres now become circles with centers at <math>(1,0)</math> and <math>(21/2,0)</math>. They have radii <math>9/2</math> and <math>4</math>, respectively. |
− | Let circle < | + | Let circle <math>A</math> be the circle centered on <math>(1,0)</math> and circle <math>B</math> be the one centered on <math>(21/2,0)</math>. |
− | The point on circle < | + | The point on circle <math>A</math> closest to the center of circle <math>B</math> is <math>(11/2,0)</math>. The point on circle B closest to the center of circle <math>A</math> is <math>(13/2,0)</math>. |
==See also== | ==See also== | ||
{{AHSME box|year=1996|num-b=26|num-a=28}} | {{AHSME box|year=1996|num-b=26|num-a=28}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:06, 30 December 2018
Contents
Problem
Consider two solid spherical balls, one centered at with radius
, and the other centered at
with radius
. How many points with only integer coordinates (lattice points) are there in the intersection of the balls?
Solution 1
The two equations of the balls are
Note that along the axis, the first ball goes from
, and the second ball goes from
. The only integer value that
can be is
.
Plugging that in to both equations, we get:
The second inequality implies the first inequality, so the only condition that matters is the second inequality.
From here, we do casework, noting that :
For , we must have
. This gives
points.
For , we can have
. This gives
points.
For , we can have
. This gives
points.
Thus, there are possible points, giving answer
.
Solution 2
Because both spheres have their centers on the x-axis, we can simplify the graph a bit by looking at a two-dimensional plane (the previous z-axis is the new x-axis and the y-axis remains the same).
The spheres now become circles with centers at and
. They have radii
and
, respectively.
Let circle
be the circle centered on
and circle
be the one centered on
.
The point on circle closest to the center of circle
is
. The point on circle B closest to the center of circle
is
.
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.