MathDB

1

Part of 2008 China Team Selection Test

Problems(7)

Chinese TST 2008 P1

Source:

4/3/2008
Let ABC ABC be a triangle, let AB>AC AB > AC. Its incircle touches side BC BC at point E E. Point D D is the second intersection of the incircle with segment AE AE (different from E E). Point F F (different from E E) is taken on segment AE AE such that CE \equal{} CF. The ray CF CF meets BD BD at point G G. Show that CF \equal{} FG.
geometrytrigonometrycircumcircleratioprojective geometrygeometry proposed
Inequality

Source: Chinese TST

4/6/2008
Let PP be an arbitrary point inside triangle ABCABC, denote by A1A_{1} (different from PP) the second intersection of line APAP with the circumcircle of triangle PBCPBC and define B1,C1B_{1},C_{1} similarly. Prove that \left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8.
inequalitiesgeometrycircumcircletrigonometryratiocyclic quadrilateralgeometry proposed
Orthocenter

Source: Chinese TST

4/4/2008
Let ABC ABC be a triangle, line l l cuts its sides BC,CA,AB BC,CA,AB at D,E,F D,E,F, respectively. Denote by O1,O2,O3 O_{1},O_{2},O_{3} the circumcenters of triangle AEF,BFD,CDE AEF,BFD,CDE, respectively. Prove that the orthocenter of triangle O1O2O3 O_{1}O_{2}O_{3} lies on line l l.
geometrycircumcirclegeometric transformationreflectiongeometry proposed
Parallel

Source: Chinese TST

4/4/2008
Let P P be the the isogonal conjugate of Q Q with respect to triangle ABC ABC, and P,Q P,Q are in the interior of triangle ABC ABC. Denote by O1,O2,O3 O_{1},O_{2},O_{3} the circumcenters of triangle PBC,PCA,PAB PBC,PCA,PAB, O1′,O2′,O3′ O'_{1},O'_{2},O'_{3} the circumcenters of triangle QBC,QCA,QAB QBC,QCA,QAB, O O the circumcenter of triangle O1O2O3 O_{1}O_{2}O_{3}, O′ O' the circumcenter of triangle O1′O2′O3′ O'_{1}O'_{2}O'_{3}. Prove that OO′ OO' is parallel to PQ PQ.
geometrycircumcircleperpendicular bisectorpower of a pointradical axisgeometry proposed
Maximum

Source: Chinese TST

4/9/2008
Given a rectangle ABCD, ABCD, let AB \equal{} b, AD \equal{} a ( a\geq b), three points X,Y,Z X,Y,Z are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by a,b a,b)
geometryrectanglegeometry proposed
Collinear

Source: Chinese TST

4/6/2008
Let ABC ABC be an acute triangle, let M,N M,N be the midpoints of minor arcs CA^,AB^ \widehat{CA},\widehat{AB} of the circumcircle of triangle ABC, ABC, point D D is the midpoint of segment MN, MN, point G G lies on minor arc BC^. \widehat{BC}. Denote by I,I1,I2 I,I_{1},I_{2} the incenters of triangle ABC,ABG,ACG ABC,ABG,ACG respectively.Let P P be the second intersection of the circumcircle of triangle GI1I2 GI_{1}I_{2} with the circumcircle of triangle ABC. ABC. Prove that three points D,I,P D,I,P are collinear.
geometrycircumcircleincentergeometric transformationreflectionratiotrigonometry
Overlapping

Source: Chinese TST

4/9/2008
Prove that in a plane, arbitrary n n points can be overlapped by discs that the sum of all the diameters is less than n n, and the distances between arbitrary two are greater than 1 1. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)
algorithminductiongeometry proposedgeometry