launched, without hitting the ceiling. This is possi
ble because the projectile does not travel along straight
lines, but instead travels along parabolic segments due
to gravity. When the projectile is at its highest point,
how high above the floor is it?
3.1.23 Recall that the ellipse is defined to be the locus
of all points in the plane, the sum of whose distance
to two fixed points (the foci) is a constant. Prove the
reflection property of the ellipse: if a pool table is built
with an elliptical wall and you shoot a ball from one
focus to any point on the wall, the ball will reflect off
the wall and travel straight to the other focus.
3.1.24 Recall that the parabola is defined to be the lo
cus of all points in the plane, such that the distance to a
fixed point (the focus) is equal to the distance to a fixed
line (the directrix). Prove the re flection property of the
parabola: if a beam of light, traveling perpendicular to
the directrix, strikes any point on the concave side of
a parabolic mirror, the beam will reflect off the mirror
and travel straight to the focus.
3.1.25 A spherical, three-dimensional planet has cen
ter at (0,0,0) and radius 20. At any point (x,y,z) on the
surface of this planet, the temperature is T(x,y,z) :=
(x + y)^2 + (y -z)^2 degrees. What is the average tem
perature of the surface of this planet?
3.1.26 (Putnam 1980) Evaluate
r/^2 dx
Jo 1 + (tanx)v'2
·
3.1.27 (Hungary 1906) Let K,L,M,N designate the
centers of the squares erected on the four sides (out
side) of a rhombus. Prove that the polygon KLMN is a
square.
3.1.28 Sharpen the problem above by showing that the
conclusion still holds if the rhombus is merely an arbi
trary parallelogram.
3.1.29 (Putnam 1992) Four points are chosen at ran
dom on the surface of a sphere. What is the probability
that the center of the sphere lies inside the tetrahedron
whose vertices are at the four points? (It is understood
3.2 The Extreme Principle
3.2 THE EXTREME PRINCIPLE 73
that each point is independently chosen relative to a
uniform distribution on the sphere.)
3.1.30 Symmetry in Probability. Imagine dropping
three pins at random on the unit interval [0, 1]. They
separate the interval into four pieces. What is the
average length of each piece? It seems obvious that
the answer "should" be 1/4, and this would be true if
the probability distributions (mean, standard deviation,
etc.) for each of the four lengths are identical. And in
deed, this is true. One way to see this is to imagine that
we are not actually dropping three pins on a line seg
ment, but instead dropping four pins on a circle with
circumference 1 unit. Wherever the fourth point lands,
cut the circle there and "unwrap" it to form the unit
interval. Ponder this argument until it makes sense.
Then try the next few problems!
(a) An ordinary deck of 52 cards with four aces is
shuffled, and then the cards are drawn one by
one until the first ace appears. On the average,
how many cards are drawn?
(b) (Jim Propp) Given a deck of 52 cards, extract 26
of the cards at random in one of the (��) possi
ble ways and place them on the top of the deck
in the same relative order as they were before
being selected. What is the expected number of
cards that now occupy the same position in the
deck as before?
(c) Given any sequence of n distinct integers, we
compute its "swap number" in the following
way: Reading from left to right, whenever we
reach a number that is less than the first num
ber in the sequence, we swap its position with
the first number in the sequence. We continue
in this way until we get to the end of the se
quence. The swap number of the sequence is
the total number of swaps. For example, the se
quence 3,4,2, I has a swap number of 2, for we
swap 3 with 2 to get 2,4, 3, 1 and then we swap
2 with 1 to get 1,4,3,2. Find the average value
of the swap numbers of the 7! = 5040 different
permutations of the integers 1,2, 3,4,5,6,7.
When you begin grappling with a problem, one of the difficulties is that there are just
so many things to keep track of and understand. A problem may involve a sequence
with many (perhaps infinitely many) elements. A geometry problem may use many
