Difference between revisions of "2020 AIME II Problems/Problem 7"
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==Problem== | ==Problem== | ||
| + | Two congruent right circular cones each with base radius <math>3</math> and height <math>8</math> have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance <math>3</math> from the base of each cone. A sphere with radius <math>r</math> lies withing both cones. The maximum possible value of <math>r^2</math> is <math>\frac{m}{n}</math>, where <math>m</math>n and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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| + | ==Solution== | ||
| + | Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, <math>O</math>, and the bases form the positive <math>x</math> and <math>y</math> axes. Then label the vertices of the region enclosed by the two triangles as <math>O,A,B,C</math> in a clockwise manner. We want to find the radius of the inscribed circle of <math>OABC</math>. By symmetry, the center of this circle must be <math>(3,3)</math>. <math>\overline{OA}</math> can be represented as <math>8x-3y=0</math> Using the point-line distance formula, <math></math>r^2=<math>\frac{15^2}{(\sqrt{8^2+3^2})^2}=\frac{225}{73}</math><math> This implies our answer is </math>225+73=\boxed{298}$. ~mn28407 | ||
| + | |||
| + | ==See Also== | ||
| + | {{AIME box|year=2020|n=II|num-b=6|num-a=8}} | ||
| + | {{MAA Notice}} | ||
| + | |||
==See Also== | ==See Also== | ||
{{AIME box|year=2020|n=II|num-b=6|num-a=8}} | {{AIME box|year=2020|n=II|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 02:30, 8 June 2020
Contents
Problem
Two congruent right circular cones each with base radius 
and height 
have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance from the base of each cone. A sphere with radius 
lies withing both cones. The maximum possible value of 
is 
, where 
n and 
are relatively prime positive integers. Find 
.
Solution
Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, 
, and the bases form the positive 
and 
axes. Then label the vertices of the region enclosed by the two triangles as 
in a clockwise manner. We want to find the radius of the inscribed circle of 
. By symmetry, the center of this circle must be 
. 
can be represented as 
Using the point-line distance formula, $$ (Error compiling LaTeX. Unknown error_msg)r^2=
225+73=\boxed{298}$. ~mn28407
See Also
| 2020 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
See Also
| 2020 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
