GeomIneq feb 26

# GeomIneq feb 26 - c = x z 6 Let a b and c be the lengths of...

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PEM Level C 26 February 2010 Geometric Inequality (Triangle Inequality) Let A , B , and C be points on the plane. Then AB + BC AC ; where equality holds if and only if B lies on the segment AC . (Side-Angle Inequality) In 4 ABC , AB > AC if and only if C > B . Exercises: 1. Let a , b , and c be the lengths of the sides of a triangle. Prove that a ( b + c - a ) < 2 bc. 2. Seven (not necessarily distinct) real numbers are taken from the open interval (1 , 13). Prove that three of them are the lengths of the sides of a triangle. 3. Let ABC be a triangle such that A < B . Prove that 1 2 c < b . 4. Let a , b , and c be positive real numbers such that a < b + c , b < a + c , and c < a + b . Prove that there exists a triangle with sides of lengths a , b , and c . 5. Prove that there exists a triangle with sides of lengths a , b , and c if and only if there exist three positive real numbers x , y , and z such that a = x + y , b = y + z , and
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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 + c-a )( c + a-b )( a + b-c ) ≤ 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 semi-perimeter. 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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