Advanced book on Mathematics Olympiad

Geometry and Trigonometry 631 0.2 0.6 0.4 0.2 0 -0.2 1 -0.4 -0.6 0 0.4 0.80.6 Figure 80 (r, θ )of the pointL. We haveAM=AL= 2 as ...

632 Geometry and Trigonometry this segment. ThenOA=cosrθandOA=ABsinθ, which yield the equation of the locus r=asinθcosθ= a 2 sin ...

Geometry and Trigonometry 633 So we want to bring the original equation of the cardioid into this form. First, we change it to r ...

634 Geometry and Trigonometry Recalling thats=x+y, we have two curves:x+y=1 and(x+y)^2 +x+y+ 1 − 3 xy=0. The last equality is eq ...

Geometry and Trigonometry 635 On the other hand, for two infinitesimally close points, the difference in the vertical tension is ...

636 Geometry and Trigonometry A glimpse at these formulas suggests the following computation: x^2 +y^2 − 1 3 z^2 =t^2 cos^2 θ+si ...

Geometry and Trigonometry 637 Figure 84 parallel cutting plane passing through the origin. Because of the conditiona>b>c,a ...

638 Geometry and Trigonometry whereis a very small positive number. Therefore, the equationf (λ)=0 has three roots,λ 1 ,λ 2 ,λ ...

Geometry and Trigonometry 639 we recognize immediately the left-hand side to be(λ 1 −λ 2 )−→v 1 ·−→v 2. We obtain the desired −→ ...

640 Geometry and Trigonometry 633.The equation of the locus can be expressed in a simple form using determinants as ∣∣ ∣∣ ∣∣ ∣ ∣ ...

Geometry and Trigonometry 641 Note thatd(X,Y) > √ 2 if and only if d^2 (X, Y )= ∑n k= 1 x^2 k+ ∑n k= 1 y^2 k− 2 ∑n k= 1 xkyk& ...

642 Geometry and Trigonometry An= ( 1 n , 1 n− 1 ·c 1 ,..., 1 3 ·cn− 3 , 1 2 ·cn− 2 ,−cn− 1 ) , An+ 1 = ( 1 n , 1 n− 1 ·c 1 ,... ...

Geometry and Trigonometry 643 Oz, andθis the oriented angle that its projection onto thexy-plane makes withOx.If we average the ...

644 Geometry and Trigonometry cost= c−xcosB √ x^2 +c^2 − 2 xccosB . The integral from the statement is ∫a 0 cost(x)dx= ∫a 0 c−xc ...

Geometry and Trigonometry 645 ξ= dx ds ,η= dy ds ,ζ= dz ds . The fact that the curve is closed simply implies that ∫L 0 ξds= ∫L ...

646 Geometry and Trigonometry Figure 87 Remark.More is true, namely that the total curvature is equal to 2πif and only if the cu ...

Geometry and Trigonometry 647 A B P C M N Figure 88 from that figure, we have A(BMP ) A(ABC) = ( BP BC ) 2 and A(CN P ) A(ABC) = ...

648 Geometry and Trigonometry 2 S 1 + 2 S 2 + 2 S 3 ≥ 1 2 (A(ABC)+S 1 +S 2 +S 3 ). The inequality follows. (M. Pimsner, S. Popa, ...

Geometry and Trigonometry 649 O NM z P α/2 α/2 Figure 91 toBC, henceAB=BCandAD=DC. Exchanging the roles ofAandCwithBand D, we fi ...

650 Geometry and Trigonometry With the substitutionc+^1 c=x, the inequality becomes x^3 − 3 x+ 1 ≤ 3 (x− 1 )^3 , forx≥ 2. But th ...