2013 UMO Problems/Problem 4
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Problem
Given line and distinct points
and
on line
, draw lines
and
through point
, with angles
and
as marked in the figure. Also, draw line segment
at an angle of
from line
such that it intersects line
at
. Establish necessary and sufficient conditions on
,
, and
such that a triangle can be drawn with one of its sides as
with lines
,
, and
as the angle bisectors of that triangle.
Solution
As we need an angle bisector, let the triangle XYI be our desired triangle. The bisected angle is clearly \beta. Using the fact that a straight line is 180 degrees, we have \alpha+2*\beta = 180 as our condition.
See Also
2013 UMO (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All UMO Problems and Solutions |