MAT102S  Introduction to Mathematical Proofs  UTM  Spring 2010
Solutions to Selected Problems from Problem Set A
1.2.
The equation
x
2
+
bx
+
c
= 0 has exactly one solution when
b
2

4
c
= 0, and it has no solutions when
b
2

4
c <
0. These statements follow from the quadratic formula (in the case where the coefficient of
x
2
is
a
= 1).
1.7.
If we take
x
= 2 and
y
=

1, then
x > y
but

1
/x
=

1
/
2 is NOT greater than

1
/y
= 1.
If we replace the condition
x > y
by
x > y >
0 then we get:
x > y
⇒
1
/x <
1
/y
⇒

1
/x >

1
/y
as required (in the first step we divided both sides by
xy
, and in the second step we multiplied both
sides by (

1)). Note
: This is not the only way to fix the false statement.
1.23.
A digital 12–hour clock broken so that the readings for minutes and for hours are always the same
can be correct every 61 minutes, except that between 12:12 and 1:01 there are only 49 minutes.
Therefore the minimum number of minutes is 49.
1.25.
We assume that the ages are positive integers. Let them be
a, b, c
with
a
≤
b
≤
c
. We are told that
abc
= 36, but that knowing
a
+
b
+
c
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 Spring '10
 Math, AGM inequality, welldefined eldest daughter

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