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Unformatted text preview: c = x + z . 6. Let a , b , and c be the lengths of the sides of a triangle. Prove that ( b + ca )( c + ab )( a + bc ) ≤ abc. When exactly does equality hold? 7. (Shortlisted Problem, IMO 1997) Let ABCDEF be a convex hexagon such that AB = BC , CD = DE , and EF = AF . Prove that BC BE + DE AD + AF CF ≥ 3 2 . When does equality occur? 8. (E¨otv¨ os 1897) Let α , β , and γ be the measures of the angles of a triangle. Prove that sin α 2 sin β 2 sin γ 2 < 1 4 . 9. In 4 ABC , let R be the circumradius, r the inradius, and s the semiperimeter. Prove that r ≤ s 3 √ 3 ≤ R 2 . 10. Let a , b , and c be the lengths of the sides of a triangle. Prove that ab + bc + ac ≤ a 2 + b 2 + c 2 < 2( ab + bc + ac ) ....
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 Spring '09
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