**UNIVERSITY OF MARYLAND MATHEMATICS COMPETITION
**
**PART II, 2001**

UNIVERSITY OF MARYLAND HIGH SCHOOL MATHEMATICS COMPETITION

PART II

December 5, 2001, 1:00--3:00

NO CALCULATORS

2 hours

1. A band of pirates unloaded some number of treasure chests
from
their ship. The number of pirates was between
60 and 69 (inclusive).
Each pirate handled exactly 11 treasure chests,
and each treasure chest was handled by exactly 7 pirates.
Exactly how many treasure chests were there? Show that your answer
is the only solution.

2. Let a and b be the lengths of the legs of a right triangle, let c
be the length of the hypotenuse, and let h be the length of the altitude
drawn from the vertex of the right angle to the hypotenuse. Prove
that c+h>a+b.

3. Prove that

1/70< (1/2)(3/4)(5/6)...(2001/2002) < 1/40.

4. Given a positive integer
a_{ 1 }
we form a sequence
a_{ 1 },
a_{ 2 },
a_{ 3 }...
as follows:
a_{ 2 }
is obtained from
a_{ 1 }
by adding together the digits of
a_{ 1 }
raised
to the 2001^{ st}
power;
a_{ 3 }
is obtained from
a_{ 2 }
using the same rule, and so on.
For example, if
a_{ 1 }=25,
then
a_{ 2 }=2^{ 2001}+5^{2001},
which is a 1399-digit number containing
106 0's,
150 1's,
124 2's,
157 3's,
148 4's,
141 5's,
128 6's,
150 7's,
152 8's,
143 9's,
So

a_{ 3 }=
106 x
0^{ 2001}+
150 x
1^{ 2001}+
124 x
2^{ 2001}+
157 x
3^{ 2001}+...+
143 x
9^{ 2001}

which is a 1912-digit number, and so forth.
Prove that if any positive integer
a_{ 1 }
is chosen to start the sequence,
then there is a positive integer M (which depends on
a_{ 1 }
) that is so large
that
a_{ n }< M for all n=1,2,3,...

5. Let P(x) be a polynomial with integer coefficients. Suppose that there are
integers a, b, and c such that P(a)=0, P(b)=1, and P(c)=2.
Prove that there is at most one integer n such that P(n)=4.