Difference between revisions of "2023 USAJMO Problems/Problem 2"
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Now we assign variables to the values of the segments. Let <math>\overline{BX}=a, \overline{XM}=b, \overline{MQ}=c, and \overline{QC}=d</math>. The equation from above gets us that <math>(a+b)c=b(c+d)</math>. As <math>a+b=c+d</math> from the problem statements, this gets us that <math>b=c</math> and <math>\overline{XC}=\overline{CQ}</math>, and we are done. | Now we assign variables to the values of the segments. Let <math>\overline{BX}=a, \overline{XM}=b, \overline{MQ}=c, and \overline{QC}=d</math>. The equation from above gets us that <math>(a+b)c=b(c+d)</math>. As <math>a+b=c+d</math> from the problem statements, this gets us that <math>b=c</math> and <math>\overline{XC}=\overline{CQ}</math>, and we are done. | ||
− | -dragoon and rhydon516 | + | -dragoon and rhydon516 (: |
Revision as of 15:14, 24 March 2023
Problem
(Holden Mui) In an acute triangle , let
be the midpoint of
. Let
be the foot of the perpendicular from
to
. Suppose that the circumcircle of triangle
intersects line
at two distinct points
and
. Let
be the midpoint of
. Prove that
.
Solution 1
The condition is solved only if is isosceles, which in turn only happens if
is perpendicular to
.
Now, draw the altitude from to
, and call that point
. Because of the Midline Theorem, the only way that this condition is met is if
, or if
.
By similarity,
. Using similarity ratios, we get that
. Rearranging, we get that
. This implies that
is cyclic.
Now we start using Power of a Point. We get that , and
from before. This leads us to get that
.
Now we assign variables to the values of the segments. Let . The equation from above gets us that
. As
from the problem statements, this gets us that
and
, and we are done.
-dragoon and rhydon516 (: