Advanced book on Mathematics Olympiad

4.1 Geometry 207 In this section we grouped problems that require only the knowledge of the theory of lines and circles. Recall ...

208 4 Geometry and Trigonometry The condition that these three points be collinear translates to 1 2 (ac−bd) ∣ ∣∣ ∣∣ ∣ ac 1 bd 1 ...

4.1 Geometry 209 590.LetMbe a point in the plane of triangleABC. Prove that the centroids of the trianglesMAB,MAC, andMCBform a ...

210 4 Geometry and Trigonometry Example.LetABCandBCDbe two equilateral triangles sharing one side. A line passing throughDinters ...

4.1 Geometry 211 We compute c−n b−m =t 2 t+ 1 +i √ 3 −t− 2 +ti √ 3 =−tei π 3 . It follows that the two lines form an angle ofπ 3 ...

212 4 Geometry and Trigonometry 603.LetA 0 ,A 1 ,...,Anbe the vertices of a regularn-gon inscribed in the unit circle. Prove tha ...

4.1 Geometry 213 through this viewpoint of one plane to another are called projective transformations. Up to a projective transf ...

214 4 Geometry and Trigonometry y 13 −y 33 +(y 1 −y 3 )(y^22 − 2 y 1 y 3 )+ 48 p^2 (y 1 −y 3 )= 0. Divide this byy 1 −y 3 =0 to ...

4.1 Geometry 215 604.Consider a circle of diameterABand centerO, and the tangenttatB. A variable tangent to the circle with cont ...

216 4 Geometry and Trigonometry inside the triangle) such that the cevians on the linesAP,BP, andCPhave equal lengths. LetSBCbe ...

4.1 Geometry 217 A few algebraic computations yield 2 y 0 t+t^2 (x−x 0 )=a(x+x 0 )+b. This shows thatxis a rational function of ...

218 4 Geometry and Trigonometry Separate the variables dx=− √ 1 −y^2 y dy, and then integrate to obtain x=− √ 1 −y^2 −lny−ln( 1 ...

4.1 Geometry 219 618.Find the locus of the projection of a fixed point on a circle onto the tangents to the circle. 619.On a cir ...

220 4 Geometry and Trigonometry x^2 −y^2 −z^2 =1, hyperboloid of two sheets; x^2 +y^2 =z, elliptic paraboloid; x^2 −y^2 =z, hyp ...

4.1 Geometry 221 yz= 2 λ x a^2 , xz= 2 λ y b^2 , yz= 2 λ z c^2 , x^2 a^2 + y^2 b^2 + z^2 c^2 = 1. Multiplying the first equation ...

222 4 Geometry and Trigonometry O y x z Figure 29 Figure 30 of the 1968 Olympic stadium in Mexico City). There is one more nonde ...

4.1 Geometry 223 628.Through a pointMon the ellipsoid x^2 a^2 + y^2 b^2 + z^2 c^2 = 1 take planes perpendicular to the axesOx,Oy ...

224 4 Geometry and Trigonometry sphere isR>0. The fixed point, which we callP, becomes the origin. The endpoints of each chor ...

4.1 Geometry 225 634.Prove that the intersection of ann-dimensional cube centered at the origin and with edges parallel to the c ...

226 4 Geometry and Trigonometry Crofton’s theorem.LetDbe a bounded convex domain in the plane. Through each pointP(x, y)outsideD ...