ProbSolv narch 6 - that the midpoint of segment PQ is the...

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PEM Level C 6 March 2010 Problem Solving 1. (IMO 1968) Prove that there is one and only one triangle whose lengths of its sides are consecutive integers, and one of whose angles is twice as large as another. 2. (IMO 1976) In a convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal. 3. (IMO 1989) Let ABCD be a convex quadrilateral such that the sides AB , AD , and BC satisfy AB = AD + BC . There exists a point P inside the quadrilateral at a distance h from CD such that AP = h + AD and BP = h + BC . Prove that 1 h 1 AD + 1 BC . 4. (IMO 2009) Let ABC be a triangle with AB = AC . The bisectors of CAB and ABC meet the sides BC and AC at D and E , respectively. Let K be the incenter of 4 ADC . Suppose that BEK = 45 . Find all possible values of CAB . Take-home Problems 5. Let 4 ABC be isosceles with AB = AC . A circle is tangent internally to the circum- circle of 4 ABC and also to the sides AB and AC at P and Q , respectively. Prove
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Unformatted text preview: that the midpoint of segment PQ is the incenter of 4 ABC . 6. (IMO 1998) A convex quadrilateral ABCD has perpendicular diagonals. The perpen-dicular bisectors of AB and CD meet at a unique point P inside ABCD . Prove that ABCD is cyclic if and only if triangles ABP and CDP have equal areas. 7. (IMO 2001) Let ABC be an acute triangle with circumcenter O and with ∠ C ≥ ∠ B + 30 ◦ . Let P be the foot of the altitude from A to BC . Prove that ∠ A + ∠ COP < 90 c irc . 8. (IMO 2009) Let ABC be a triangle with circumcenter O . The points P and Q are interior points of sides AC and AB , respectively. Let K , L , and M be midpoints of the segments BP , CQ , and PQ , respectively, and let Σ be the circle passing through K , L , and M . Suppose that the line PQ is tangent to the circle Σ Prove that OP = OQ ....
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