132 CHAPTER 4 THREE IMPORTANT CROSSOVER TACTICSchange interpretation. Recall that mUltiplica
tion by i means "rotate by 90 degrees counter
clockwise." Show that this implies that f(t) is a
circular path.
(c) Think about the speed at which this circular
curve t >--> f (t) is being traced out, and concl ude
that eit = cost + isint.
4.2.30 Let Ra (e) denote the transformation of the
plane that rotates everything about the center point a
by e radians counterclockwise. Prove the interesting
fact that the composition of Ra(e) and Rb(l/J) is an
other rotation Rc( a). Find c, a in terms of a,h, e, I/J.
Does this agree with your intuition?
4.2.31 Show that there do not exist any equilateral tri
angles in the plane whose vertices are lattice points
(integer coordinates).
4.2.32 Show that the triangle with vertices a,b,c in
the complex plane is equilateral if and only if
a^2 +h^2 +c^2 = ah+hc+ca.
4.2.33 Find necessary and sufficient conditions for the
two roots of z^2 + az + h = 0, plus 0, to form the vertices
of an equilateral triangle.
4.2.34 (T. Needham) Draw any quadrilateral, and on
each side draw a square lying outside the given quadri
lateral. Draw line segments joining the centers of op
posite squares. Show that these two line segments are
perpendicular and equal in length.
4.2.35 (T. Needham) Let ABC be a triangle, with
points P, Q, R situated outside ABC so that triangles
PAC, RCB, QBA are similar to one another. Then the
centroids (intersection of medians) of ABC and PQR
are the same point.
4.2.36 (T. Needham) Draw any triangle, and on each
side draw an equilateral triangle lying outside the
given triangle. Show that the centroids of these three
equilateral triangles are the vertices of an equilateral
triangle. (The centroid of a triangle is the intersection
of its medians; it is also the center of gravity.) If the4.3 Generating Functions
verti ces of a triangle are the complex numbers a,h,c,
the centroid is located at (a+h+c)/3.
4.2.37 (Hungary 1941) Hexagon ABCDEF is in
scribed in a circle. The sides AB, CD, and EF are all
equal in length to the radius. Prove that the midpoints
of the other three sides are the vertices of an equilateral
triangle.
4.2.38 (lMO proposal) Let n be a positive integer hav
ing at least two distinct prime factors. Show that there
is a permutation (al ,a 2 , ... ,an) of (l,2, ... ,n) such
that± kcos
2 nak
=
o.
k= 1 n
4.2.39 For each positive integer n, define the polyno
mialDo the zeros of Pn lie inside, outside, or on the unit
circle Izl = I?
4.2.40 (Putnam 1998) Let s be any arc of the unit cir
cle lying entirely in the first quadrant. Let A be the
area of the region lying below s and above the x-axis
and let B be the area of the region lying to the right of
the y-axis and to the left of s. Prove that A + B depends
only on the arc length, and not on the position, of s.
4.2.41 Ptolemy's Theorem. Let a,h,c ,d be four arbi
trary complex numbers. Verify the identity
(a-c)(h -d) = (a- c)(c-d) +(h-c) (a-d).
Apply the triangle inequality to deduce Ptolemy's In
equality from plane geometry (not involving complex
numbers): For any four points A,B,C,D in the plane,
AC ·BD "5. AB ·CD +BC· DA ,
with equality if and only if the quadrilateral ABCD is
convex and cyclic (i.e., is inscribed in a circle). The
equality part, more commonly called Ptolemy's The
orem, is the trickiest part of this problem. For otherways to prove this, see Problems 8.4.30 and (^8) .5.49.
The crossover tactic of generating functions owes its power to two simple facts.
- When you multiply xn by Xl , you get xn+n.