C10.Find the number of rotationally distinct ways of coloring a tetrahedron’s edgeswith three distinct colors.(A)80(B)87(C)96(D)105(E)1154
(Karen Ge)3 Geometry3 GeometryG1.(2011A) TriangleABChas∠BAC= 60◦,∠CBA≤90◦,BC= 1, andAC≥AB. LetH,I, andObe the orthocenter, incenter, and circumcenter of4ABC,respectively. Assume that the area of pentagonBCOIHis the maximum possible.What is∠CBA?4ABChas a right angle atC. PointPis inside4ABC, suchthatPA= 11,PB= 7, andPC= 6. LegsACandBChave lengths=pa+b√2,whereaandbare positive integers. What isa+b?G3.(2007B) PointsA, B, C, DandEare located in 3-dimensional space withAB=BC=CD=DE=EA= 2 and∠ABC=∠CDE=∠DEA= 90o. The plane of4ABCis parallel toDE. What is the area of4BDE?G4.(2008B) LetABCDbe a trapezoid withAB||CD,AB= 11,BC= 5,CD= 19,andDA= 7. Bisectors of∠Aand∠Dmeet atP, and bisectors of∠Band∠Cmeet atQ. What is the area of hexagonABQCDP?(A)28√3(B)30√3(C)32√3(D)35√3(E)36√3G5.(2012B) SquareAXY Zis inscribed in equiangular hexagonABCDEFwithXonBC,YonDE, andZonEF. Suppose thatAB= 40, andEF= 41(√3-1).What is the side-length of the square?√2 +41√3(C)20√3 + 16G6.(2002B) A convex quadrilateralABCDwith area 2002 contains a pointPin itsinterior such thatPA= 24, PB= 32, PC= 28, PD= 45. Find the perimeter ofABCD.113)G7.(2015B) Four circles, no two of which are congruent, have centers atA,B,C, andD, and pointsPandQlie on all four circles. The radius of circleAis58timesthe radius of circleB, and the radius of circleCis58times the radius of circleD.Furthermore,AB=CD= 39 andPQ= 48. LetRbe the midpoint ofPQ. WhatisAR+BR+CR+DR?5
(Karen Ge)3 GeometryG8.(2015A) A collection of circles in the upper half-plane, all tangent to thex-axis,is constructed in layers as follows. LayerL0consists of two circles of radii 702and 732that are externally tangent. Fork≥1, the circles inSk-1j=0Ljare orderedaccording to their points of tangency with thex-axis. For every pair of consecutivecircles in this order, a new circle is constructed externally tangent to each of thetwo circles in the pair. LayerLkconsists of the 2k-1circles constructed in this way.LetS=S6j=0Lj, and for every circleCdenote byr(C) its radius. What isX1pr(C)?BPAP-APBPgiven thatQPAP+APQP=51144-2.