Difference between revisions of "2001 IMO Problems/Problem 1"
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Notice that because <math>\angle{PCO} = 90^\circ - \angle{A}</math>, it suffices to prove that <math>\angle{POC} < \angle{PCO}</math>, or equivalently <math>PC < PO.</math> | Notice that because <math>\angle{PCO} = 90^\circ - \angle{A}</math>, it suffices to prove that <math>\angle{POC} < \angle{PCO}</math>, or equivalently <math>PC < PO.</math> | ||
− | Suppose on the contrary that <math>PC > PO</math>. By the triangle inequality, <math>2 PC = PC + PC > PC + PO | + | Suppose on the contrary that <math>PC > PO</math>. By the triangle inequality, <math>2 PC = PC + PC > PC + PO > CO = R</math>, where <math>R</math> is the circumradius of <math>ABC</math>. But the Law of Sines and basic trigonometry gives us that <math>PC = 2R \sin B \cos C</math>, so we have <math>4 \sin B \cos C > 1</math>. But we also have <math>4 \sin B \cos C \le 4 \sin B \cos (B + 30^\circ) = 2 (\sin (2B + 30^\circ) - \sin 30^\circ) \le 2 (1 - \frac{1}{2}) = 1</math> because <math>\angle{C} \ge \angle{B} + 30^\circ</math>, and so we have a contradiction. Hence <math>PC < PO</math> and so <math>\angle{PCO} + \angle{A} < 90^\circ</math>, as desired. |
== See also == | == See also == |
Revision as of 16:13, 21 July 2018
Contents
[hide]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
Take on the circumcircle with . Notice that , so . Hence . Let be the midpoint of and the midpoint of . Then , where is the radius of the circumcircle. But (since is a rectangle).
Now cannot coincide with (otherwise would be and the triangle would not be acute-angled). So . But . So .
Hence . Let be a diameter of the circle, so that . But and , since is a diameter. Hence .
Solution 2
Notice that because , it suffices to prove that , or equivalently
Suppose on the contrary that . By the triangle inequality, , where is the circumradius of . But the Law of Sines and basic trigonometry gives us that , so we have . But we also have because , and so we have a contradiction. Hence and so , as desired.
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 |