MA017 2010 HOMEWORK 2
(1)
Write
n
ones on the board, and repeat the process “erase any two numbers
a,b
and write
a
+
b
4
” until
one number
x
remains. Show that
x
≥
1
/n
.
Solution.
The sum
∑
1
/a
≤
n
, taken over the numbers
a
on the board, is nonincreasing: for any
two positive numbers
a,b
, by rearranging (
a

b
)
2
≥
0 we ﬁnd that
4
a
+
b
≤
1
/a
+ 1
/b.
Thus 1
/x
≤
1
/
1 +
···
+ 1
/
1 =
n
.
(2)
Given ﬁnitely many points in the plane, any three of which form a triangle of area at most one, show
that all of them lie inside some triangle of area four, including its boundary.
Solution.
Amongst the ﬁnitely many triangles formed by three such points, choose
ABC
of maximal
area. Let
a,b,c
be the lines parallel to
BC,CA,AB
passing through
A,B,C
, respectively. They
bound a triangle Δ of area at most four. If any point
P
in our set does not lie in Δ, it lies on the
other side of
a,b
or
c
, and we may assume without loss of generality that it lies on the other side of
a
. Then
P
is farther from the line
BC
than
A
is, so
PBC
has greater area than
ABC
, contradiction.
(3)
Starting with the number
7
2008
, cross out the ﬁrst digit and add it to the remaining number (e.g.,
325
7→
28
). You repeat this until a tendigit number
x
remains. Show that
x
has two equal digits.
Solution.
The invariant is
n
(mod 9): if
n
has
e
digits and ﬁrst digit
n
0
, then we replace it with
n

10
e
n
0
+
n
0
=
n

(10
e

1)
n
0
≡
n
(mod 9). If
x
does not have two equal digits, its digits are a
permutation of 0
,
1
,
2
,...,
9, so