Advanced book on Mathematics Olympiad

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 , as desired. 

The second example comes from the 15th W.L. Putnam Mathematical Competition,
1955.

Example.LetA 1 A 2 A 3 ...Anbe a regular polygon inscribed in the circle of centerOand
radiusr. On the half-line|OA 1 choose the pointPsuch thatA 1 is betweenOandP.
Prove that

∏n

i= 1

PAi=POn−rn.

Solution.Place the vertices in the complex plane such thatAi=ri,1≤i≤n, where
is annth root of unity. The coordinate ofPis a real numberrx, withx>1. We have

∏n

i= 1

PAi=

∏n

i= 1

|rx−ri|=rn

∏n

i= 1

|x−i|=rn

∣∣

∣∣

∣

∏n

i= 1

(x−i)

∣∣

∣∣

∣

=rn(xn− 1 )=(rx)n−rn=POn−rn.

The identity is proved. 

598.LetABCDEFbe a hexagon inscribed in a circle of radiusr. Show that ifAB=
CD=EF=r, then the midpoints ofBC,DE, andFAare the vertices of an
equilateral triangle.

599.Prove that in a triangle the orthocenterH, centroidG, and circumcenterOare
collinear. Moreover,Glies betweenHandO, andOGGH=^12.

600.On the sides of a convex quadrilateralABCDone draws outside the equilateral
trianglesABMandCDPand inside the equilateral trianglesBCNandADQ.
Describe the shape of the quadrilateralMNPQ.

601.LetABCbe a triangle. The trianglesPABandQACare constructed outside of
the triangleABCsuch thatAP=AB,AQ=AC, and∠BAP=∠CAQ=α.
The segmentsBQandCPmeet atR. LetObe the circumcenter of the triangle
BCR. Prove thatAOandPQare orthogonal.

602.LetA 1 A 2 ...Anbe a regular polygon with circumradius equal to 1. Find the max-
imum value of

∏n

k= 1 PAkasPranges over the circumcircle.