100 Geometry Problems: Bridging the Gap from AIME to …

1 100 geometry problems : Bridging the Gap from AIME to USAMOD avid AltizioAugust 30, 2014 AbstractThis is a collection of one-hundred Geometry problems from all around the globe designed for Bridging thegap between computational Geometry and proof Geometry . Problems start middle-AMC level and go all the wayto early IMO Shortlist level. As there are computational and proof problems mixed in with each other, relativedifficulties may not be exact, so feel free to skip around. Enjoy!11. [MA ????] In the figure shown below, circleBis tangent to circleAatX, circleCis tangent to circleAatY, and circlesBandCare tangent to each other.

2 IfAB= 6,AC= 5, andBC= 9, what isAX?ABCYX2. [AHSME ????] In triangleABC,AC=CDand CAB ABC= 30 . What is the measure of BAD?ABCD3. [AMC 10A 2004] SquareABCDhas side length 2. A semicircle with diameterABis constructed inside thesquare, and the tangent to the semicircle fromCintersects sideADatE. What is the length ofCE?ABCDE4. [AMC 10B 2011] RectangleABCDhasAB= 6 andBC= 3. PointMis chosen on sideABso that AMD= CMD. What is the degree measure of AMD?1 This is the second version of the PDF. It fixes a few typoes and inaccuracies found in the first Geometry ProblemsDavid AltizioPage 25.

3 [AIME 2011] On squareABCD, pointElies on sideADand pointFlies on sideBC, so thatBE=EF=FD= 30. Find the area of the PointsA, B,andCare situated in the plane such that ABC= 90 . LetDbe an arbitrary point onAB,and letEbe the foot of the perpendicular fromDtoAC. Prove that DBE= [AMC 10B 2012] Four distinct points are arranged in a plane so that the segments connecting them havelengthsa, a, a, a,2a,andb. What is the ratio ofbtoa?8. [Britain 2010] LetABCbe a triangle with CABa right angle. The pointLlies on the sideBCbetweenBandC. The circleBALmeets the lineACagain atMand the circleCALmeets the lineABagain thatL,M, andNlie on a straight [OMO 2014] LetABCbe a triangle with incenterIandAB= 1400,AC= 1800,BC= 2014.

4 The circlecentered atIpassing throughAintersects lineBCat two pointsXandY. Compute the [India RMO 2014] LetABCbe an isosceles triangle withAB=ACand let denote its circumcircle. A pointDis on arcABof not containingC. A pointEis on arcACof not containingB. IfAD=CEprovethatBEis parallel A closed planar shape is said to beequiableif the numerical values of its perimeter and area are the example, a square with side length 4 is equiable since its perimeter and area are both 16. Show that anyclosed shape in the plane can be dilated to become equiable. (A dilation is an affine transformation in whicha shape is stretched or shrunk.)

5 In other words, ifAis a dilated version ofBthenAis similar toB.)12. [David Altizio] TriangleAEFis a right triangle withAE= 4 andEF= 3. The triangle is inscribed insidesquareABCDas shown. What is the area of the square?ABCDEF13. PointsAandBare located on circle , and pointCis an arbitrary point in the interior of . ExtendACandBCpastCso that they hit atMandNrespectively. LetXdenote the foot of the perpendicular fromMtoBN, and letYdenote the foot of the perpendicular fromNtoAM. Prove thatAB [AIME 2007] SquareABCDhas side length 13, and pointsEandFare exterior to the square such thatBE=DF= 5 andAE=CF= 12.

6 Let be the circumcircle of4 ABC, and letD, E, Fbe the midpoints of arcsAB, BC, CArespectively. ProvethatDF [AIME 1984] In tetrahedronABCD, edgeABhas length 3 cm. The area of faceABCis 15 cm2and the areaof faceABDis 12 cm2. These two faces meet each other at a 30 angle. Find the volume of the tetrahedronin LetP1P2P3P4be a quadrilateral inscribed in a circle with diameter of lengthD, and letXbe the intersectionof its diagonals. IfP1P3 P2P4prove thatD2=XP21+XP22+XP23+ [iTest 2008] Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points,AandB, such thatAB= 42.

7 If the radii of the two circles are 54 and 66, findR2, whereRis the radius ofthe Geometry ProblemsDavid AltizioPage 319. [AIME 2008] In trapezoidABCD withBC AD, letBC= 1000 andAD= 2008. Let A= 37 , D= 53 ,andMandNbe the midpoints ofBCandAD, respectively. Find the [Sharygin 2014] LetABCbe an isosceles triangle with baseAB. Line`touches its circumcircle at a perpendicular fromCto`, andAE,BFbe the altitudes ofABC. Prove thatD, E, [Purple Comet 2013] Two concentric circles have radii 1 and 4. Six congruent circles form a ring where eachof the six circles is tangent to the two circles adjacent to it as shown.

8 The three lightly shaded circles areinternally tangent to the circle with radius 4 while the three darkly shaded circles are externally tangent tothe circle with radius 1. The radius of the six congruent circles can be writtenk+ mn, wherek, m,andnareintegers withkandnrelatively prime. Findk+m+ LetA,B,C, andDbe points in the plane such that BAC= CBD. Prove that the circumcircle of4 ABCis tangent [Britain 1995] TriangleABChas a right angle atC. The internal bisectors of anglesBACandABCmeetBCandCAatPandQrespect ively. The pointsMandNare the feet of the perpendiculars fromPandQtoAB. Find LetABCDbe a parallelogram with Aobtuse, and letMandNbe the feet of the perpendiculars fromAto sidesBCandCD.

9 Prove that4 MAN For a given triangle4 ABC, letHdenote its orthocenter andOits circumcenter.(a) Prove that HAB= (b) Prove that HAO=| B C|.26. SupposeP, A, B, C,andDare points in the plane such that4 PAB 4 PCD. Prove that4 PAC [AMC 12A 2012] CircleC1has its centerOlying on circleC2. The two circles meet atXandY. PointZinthe exterior ofC1lies on circleC2andXZ= 13,OZ= 11, andY Z= 7. What is the radius of circleC1?28. LetABCDbe a cyclic quadrilateral with no two sides parallel. LinesADandBC(extended) meet atK,andABandCD(extended) meet angle bisector of DKCintersectsCDandABat pointsEandF,respectively; the angle bisector of CMBintersectsBCandADat pointsGandH,respectively.

10 Prove thatquadrilateralEGFHis a [David Altizio] In4 ABC,AB= 13,AC= 14, andBC= 15. LetMdenote the midpoint ofAC. PointPis placed on line segmentBMsuch thatAP PC. Suppose thatp, q,andrare positive integers withpandrrelatively prime andqsquarefree such that the area of4 APCcan be written in the formp qr. What isp+q+r?30. [All-Russian MO 2013] Acute-angled triangleABCis inscribed into circle . Lines tangent to atBandCintersect atP. PointsDandEare onABandACsuch thatPDandPEare perpendicular toABandACrespectively. Prove that the orthocenter of triangleADEis the midpoint a result of this equality condition, linesAHandAOare said to beisogonal conjugates, reflections across theA-angle Geometry ProblemsDavid AltizioPage 431.