Difference between revisions of "2019 CIME I Problems/Problem 14"
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Let <math>ABC</math> be a triangle with circumcenter <math>O</math> and incenter <math>I</math> such that the lengths of the three segments <math>AB</math>, <math>BC</math>, and <math>CA</math> form an increasing arithmetic progression in this order. If <math>AO=60</math> and <math>AI=58</math>, then the distance from <math>A</math> to <math>BC</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Let <math>ABC</math> be a triangle with circumcenter <math>O</math> and incenter <math>I</math> such that the lengths of the three segments <math>AB</math>, <math>BC</math>, and <math>CA</math> form an increasing arithmetic progression in this order. If <math>AO=60</math> and <math>AI=58</math>, then the distance from <math>A</math> to <math>BC</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
| + | ==Video Solution by MOP 2024== | ||
| + | https://youtube.com/watch?v=hIYTLWi1HW0 | ||
| + | |||
| + | ~r00tsOfUnity | ||
| + | |||
| + | ==See also== | ||
{{CIME box|year=2019|n=I|num-b=13|num-a=15}} | {{CIME box|year=2019|n=I|num-b=13|num-a=15}} | ||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
{{MAC Notice}} | {{MAC Notice}} | ||
Latest revision as of 19:51, 15 January 2024
Let 
be a triangle with circumcenter 
and incenter 
such that the lengths of the three segments 
, 
, and 
form an increasing arithmetic progression in this order. If 
and 
, then the distance from 
to can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Video Solution by MOP 2024
https://youtube.com/watch?v=hIYTLWi1HW0
~r00tsOfUnity
See also
| 2019 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All CIME Problems and Solutions | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
