Advanced book on Mathematics Olympiad

Geometry and Trigonometry 611 Since −→ AB× −→ ACand −→ AE× −→ ADare perpendicular to the plane of the triangle and oriented the ...

612 Geometry and Trigonometry Arguing similarly for the other two inequalities, we deduce that the locus is the interior of the ...

Geometry and Trigonometry 613 A M B O C x y P Figure 74 HenceChas coordinates( 0 , 2 cbc−a). The slope of the lineCMisba, so the ...

614 Geometry and Trigonometry A B D C H G E F Figure 75 LetEbe the origin of the rectangular system of coordinates, with lineEBa ...

Geometry and Trigonometry 615 Let us rephrase this in geometric terms. We are required to include a segmenty=px+q, 0 ≤x≤1, betwe ...

616 Geometry and Trigonometry = ( − 1 x 1 x 2 x 3 x 4 + 1 x^21 x 22 x^23 x 42 ) ∣ ∣∣ ∣∣ ∣∣ ∣ x^31 x 12 x 11 x^32 x 22 x 21 x^33 ...

Geometry and Trigonometry 617 m−a b−a = n−c b−c = p−c d−c = q−a d−a =, where=cosπ 3 +isinπ 3. Therefore, m=a+(b−a), n=c+(b−c) ...

618 Geometry and Trigonometry On the other hand,Pis obtained by rotatingBaroundAby−α, so its coordinate is p =bω ̄. Similarly, t ...

Geometry and Trigonometry 619 Remark.Let us show how this geometric identity can be used to derive a trigono- metric identity. F ...

620 Geometry and Trigonometry thereforeQP=OQ. We conclude thatQlies on the parabola of focusOand directrix t. A continuity argum ...

Geometry and Trigonometry 621 607.Since the property we are trying to prove is invariant under affine changes of coor- dinates, ...

622 Geometry and Trigonometry m 1 +m 2 = y 0 x 0 , m 1 m 2 = p x 0 . We also know that the angle between the tangents isφ. We di ...

Geometry and Trigonometry 623 ± 1 2 ∣∣ ∣∣ ∣ ∣ 4 pα^214 pα 11 4 pα^224 pα 21 4 pα^234 pα 31 ∣∣ ∣∣ ∣ ∣ =± 8 p^2 ∣∣ ∣∣ ∣ ∣ α^21 α 1 ...

624 Geometry and Trigonometry (c) Substituting the coordinates ofLin the equation of the circle yields (ac+ 4 p^2 )(a−b)(c−b)= 0 ...

Geometry and Trigonometry 625 x^2 +y^2 +cx+dy+e= 0 ,i= 1 , 2 ,...,n. Writing the fact that the vertices also satisfy the equatio ...

626 Geometry and Trigonometry =n− ∑n k= 1 cos ( α+ 2 (k− 1 )π n ) =n−cosα ∑n k= 1 cos ( 2 (k− 1 )π n ) +sinα ∑n k= 1 sin ( 2 (k− ...

Geometry and Trigonometry 627 P= k+ 2 r 3 k A+ 2 k− 2 r 3 k D = k−r−3 cosθ 3 k B+ 2 k+r+3 cosθ 3 k E = k−r+3 cosθ 3 k C+ 2 k+r−3 ...

628 Geometry and Trigonometry The substitutiont=tanu 2 changes this into 2 a+d ∫ dt 1 +aa−+ddt^2 . Ifa=dthe answer to the proble ...

Geometry and Trigonometry 629 X(t^2 X−tX−t+ 2 )= 0. The rootX=0 corresponds to the origin. The other rootX=tt 2 −−^2 t gives the ...

630 Geometry and Trigonometry 01 0.2 40 0.8 20 (^0) 0.6 -20 0.4 -40 Figure 79 with the equation of the cardioid, we find the pos ...