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Turkey NMO 2008 1st Round - P05 (Geometry)

Source:

August 24, 2012
geometryPythagorean Theorem

Problem Statement

A triangle with sides a,b,ca,b,c is called a good triangle if a2,b2,c2a^2,b^2,c^2 can form a triangle. How many of below triangles are good?
(i) 40,60,8040^{\circ}, 60^{\circ}, 80^{\circ}
(ii) 10,10,16010^{\circ}, 10^{\circ}, 160^{\circ}
(iii) 110,35,35110^{\circ}, 35^{\circ}, 35^{\circ}
(iv) 50,30,10050^{\circ}, 30^{\circ}, 100^{\circ}
(v) 90,40,5090^{\circ}, 40^{\circ}, 50^{\circ}
(vi) 80,20,8080^{\circ}, 20^{\circ}, 80^{\circ}
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> 5 <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5