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MATH 135
Fall 2008
Assignment #1
Due: Wednesday 17 September 2008, 8:20 a.m.
N.B. Assignments 3 to 9 will not be distributed in class. You must download them from the course
Web site.
HandIn Problems
1. Disprove the statement “There is no positive integer
n >
3 such that
n
2
+ (
n
+ 1)
2
is a perfect
square”.
2. (a) Treating the equation 4
x
2
+ 4
xy
+ 2
y
2

4
x

2
y
+ 1 = 0 as a quadratic equation in
x
with
coeﬃcients in terms of
y
, solve for
x
.
(b) Disprove the statement “For all real numbers
x
and
y
, 4
x
2
+4
xy
+2
y
2

4
x

2
y
+1
6
= 0”.
3. Prove that sin
4
x
+ 2 cos
2
x

cos
4
x
= 1 for all
x
.
4. Prove that if 2
≤
x
≤
5, then

2
≤
x
2

3
x
≤
10.
5. Any positive integer
n
with units digit 5 can be written in the form
n
= 10
k
+ 5, for some
nonnegative integer
k
. Prove that if
n
is a positive integer with units digit 5, then
n
2
ends
with 25.
6. Prove that if
n
is a positive integer with
n
≥
2, then
n
3

n
is always divisible by 6.
7. To prove a statement of the form “
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 Fall '08
 ANDREWCHILDS
 Math, Algebra

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