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Michigan Mathematics Prize Competition
1967 MMPC
1967 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1967 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Consider the system of simultaneous equations
(
x
+
y
)
(
x
+
z
)
=
a
2
(x+y)(x+z)=a^2
(
x
+
y
)
(
x
+
z
)
=
a
2
(
x
+
y
)
(
y
+
z
)
=
b
2
(x+y)(y+z)=b^2
(
x
+
y
)
(
y
+
z
)
=
b
2
(
x
+
z
)
(
y
+
z
)
=
c
2
(x+z)(y+z)=c^2
(
x
+
z
)
(
y
+
z
)
=
c
2
, where
a
b
c
≠
0
abc \ne 0
ab
c
=
0
. Find all solutions
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in terms of
a
a
a
,
b
b
b
, and
c
c
c
. p2. Shown in the figure is a triangle
P
Q
R
PQR
PQR
upon whose sides squares of areas
13
13
13
,
25
25
25
, and
36
36
36
sq. units have been constructed. Find the area of the hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
. https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png p3. Suppose
p
,
q
p,q
p
,
q
, and
r
r
r
are positive integers no two of which have a common factor larger than
1
1
1
. Suppose
P
,
Q
P,Q
P
,
Q
, and
R
R
R
are positive integers such that
P
p
+
Q
q
+
R
r
\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}
p
P
+
q
Q
+
r
R
is an integer. Prove that each of
P
/
p
P/p
P
/
p
,
Q
/
q
Q/q
Q
/
q
, and
R
/
r
R/r
R
/
r
is an integer. p4. An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png p5. Suppose that
p
1
p_1
p
1
,
p
2
p_2
p
2
,
p
3
p_3
p
3
and
p
4
p_4
p
4
are the centers of four non-overlapping circles of radius
1
1
1
in a plane and that,
p
p
p
is any point in that plane. Prove that
p
1
p
‾
2
+
p
2
p
‾
2
+
p
3
p
‾
2
+
p
4
p
‾
2
≥
6.
\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.
p
1
p
2
+
p
2
p
2
+
p
3
p
2
+
p
4
p
2
≥
6.
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