Difference between revisions of "2001 IMO Problems/Problem 1"
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| − | ==Problem== | + | == Problem == |
Consider an acute triangle <math>\triangle ABC</math>. Let <math>P</math> be the foot of the altitude of triangle <math>\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assume that <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. | Consider an acute triangle <math>\triangle ABC</math>. Let <math>P</math> be the foot of the altitude of triangle <math>\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assume that <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. | ||
| − | ==Solution== | + | == Solution == |
{{solution}} | {{solution}} | ||
| + | == See also == | ||
{{IMO box|year=2001|before=First question|num-a=2}} | {{IMO box|year=2001|before=First question|num-a=2}} | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
Revision as of 20:58, 21 April 2008
Problem
Consider an acute triangle 
. Let 
be the foot of the altitude of triangle issuing from the vertex 
, and let 
be the circumcenter of triangle . Assume that 
. Prove that 
.
Solution
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See also
| 2001 IMO (Problems) • Resources | ||
| Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
