The Number Games Sprint Math Worksheet - College Of Charleston, 2014
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1
2The Number Games Sprint
College of Charleston Math Meet
February 22, 2014
Instructions:
Write the exact answer to each question in the corresponding box. Remember that the
winners in this event are those participants who answer the most questions correctly in a row beginning with
the first question. So, try to get as far as you can without making a mistake!
2014
x
1. Suppose
and
are nonzero real numbers satisfying
. Find the exact value of
.
x
y
x + y = 0
2014
y
# 1:
2014
2. What are the last two digits of
?
11
# 2:
2
2
3. Suppose
and
are two real numbers such that
and
. Find the value of
.
x − y = 3
x
y
x
+ y
= 4037
xy
# 3:
√
√
√
4. Suppose
and
are two real numbers such that
and
. Find the value of
.
x
y
x +
y = 50
x + y = 2014
xy
# 4:
5. Every integer has a unique prime factorization. For example, the prime factorization of
is
2
60
2
× 3 × 5
3
1007
2
1007
and the prime factorization of
is
. Let
and let
+ · · · + c
27
3
p(x) = (x + 3)
= c
+ c
x + c
x
x
0
1
2
1007
be the sum of the coefficients from its expanded form. What is the prime
+ · · · + c
N = c
+ c
+ c
+ c
0
1
2
1006
1007
factorization of
?
N
# 5:
6. Recall that
is the factorial of
.
Determine the remainder when
n! = n(n − 1)(n − 2) · · · (3)(2)(1)
n
2
is divided by
.
(1! + 2! + · · · + 2014!)
4
# 6:
7.
ends in a lot of zeros. How many are there?
2014!
# 7:
8. Express the following product as a fraction in lowest terms:
1
1
1
1
2 · 1 −
1 −
1 −
· · · 1 −
2
2
3
2
4
2
2014
2
# 8:
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