Difference between revisions of "2001 IMO Problems"

Revision as of 01:52, 6 October 2014

Problem 1

Consider an acute triangle $\triangle ABC$ . Let $P$ be the foot of the altitude of triangle $\triangle ABC$ issuing from the vertex $A$ , and let $O$ be the circumcenter of triangle $\triangle ABC$ . Assume that $\angle C \geq \angle B+30^{\circ}$ . Prove that $\angle A+\angle COP < 90^{\circ}$ .

Problem 2

Let $a,b,c$ be positive real numbers. Prove that $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$ .

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 ==Problem 2==
+Let <math>a,b,c</math> be positive real numbers. Prove that <math>\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1</math>.
+__TOC__
 ==Problem 3==

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Difference between revisions of "2001 IMO Problems"

Revision as of 01:52, 6 October 2014

Problem 1

Problem 2

Contents

Problem 3

Problem 4

Problem 5

Problem 6

See Also