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 14:14, 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 (: