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1993 MMPC
1993 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1993 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. A matrix is a rectangular array of numbers. For example,
(
1
2
3
4
)
\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
(
1
3
2
4
)
and
(
1
3
2
4
)
\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}
(
1
2
3
4
)
are
2
×
2
2 \times 2
2
×
2
matrices. A saddle point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a
2
×
2
2 \times 2
2
×
2
matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a
2
×
2
2 \times 2
2
×
2
matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. p2. a. Find four different pairs of positive integers satisfying the equation
7
m
+
11
n
=
1
\frac{7}{m}+\frac{11}{n}=1
m
7
+
n
11
=
1
. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation
7
m
+
11
n
=
1
\frac{7}{m}+\frac{11}{n}=1
m
7
+
n
11
=
1
. p3. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral, and let points
M
,
N
,
O
,
P
M, N, O, P
M
,
N
,
O
,
P
be the respective midpoints of sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
. a. Show, by example, that it is possible that
A
B
C
D
ABCD
A
BC
D
is not a parallelogram, but
M
N
O
P
MNOP
MNOP
is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that
M
O
MO
MO
is perpendicular to
N
P
NP
NP
. Prove that
A
C
=
B
D
AC = BD
A
C
=
B
D
. p4. A Pythagorean triple is an ordered collection of three positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
satisfying the relation
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
. We say that
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is a primitive Pythagorean triple if
1
1
1
is the only common factor of
a
,
b
a, b
a
,
b
, and
c
c
c
. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
where
c
=
b
+
2
c = b + 2
c
=
b
+
2
. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
where
c
=
b
+
3
c = b + 3
c
=
b
+
3
. p5. Let
x
x
x
and
y
y
y
be positive real numbers and let
s
s
s
be the smallest among the numbers
3
x
2
\frac{3x}{2}
2
3
x
,
y
x
+
1
x
\frac{y}{x}+\frac{1}{x}
x
y
+
x
1
and
3
y
\frac{3}{y}
y
3
. a. Find an example giving
s
>
1
s > 1
s
>
1
. b. Prove that for any positive
x
x
x
and
y
,
s
<
2
y,s <2
y
,
s
<
2
. c. Find, with proof, the largest possible value of
s
s
s
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.