Consider an acute-angled 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 .
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Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
Source: Međunarodna matematička olimpijada, shortlist 2001
Školjka 
. Let 
be the foot of the altitude of triangle 
, and let 
be the circumcenter of triangle 
. Prove that 
.