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Michigan Mathematics Prize Competition
1979 MMPC
1979 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1979 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Solve for
x
x
x
and
y
y
y
if
1
x
2
+
1
x
y
=
1
9
\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}
x
2
1
+
x
y
1
=
9
1
and
1
y
2
+
1
x
y
=
1
16
\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}
y
2
1
+
x
y
1
=
16
1
p2. Find positive integers
p
p
p
and
q
q
q
, with
q
q
q
as small as possible, such that
7
10
<
p
q
<
11
15
\frac{7}{10} <\frac{p}{q} <\frac{11}{15}
10
7
<
q
p
<
15
11
. p3. Define
a
1
=
2
a_1 = 2
a
1
=
2
and
a
n
+
1
=
a
n
2
−
a
n
+
1
a_{n+1} = a^2_n -a_n + 1
a
n
+
1
=
a
n
2
−
a
n
+
1
for all positive integers
n
n
n
. If
i
>
j
i > j
i
>
j
, prove that
a
i
a_i
a
i
and
a
j
a_j
a
j
have no common prime factor. p4. A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. p5. In triangle
A
B
C
ABC
A
BC
, let
∠
A
B
C
=
∠
A
C
B
=
4
0
o
\angle ABC=\angle ACB=40^o
∠
A
BC
=
∠
A
CB
=
4
0
o
is extended to
D
D
D
such that
A
D
=
B
C
AD=BC
A
D
=
BC
. Prove that
∠
B
C
D
=
1
0
o
\angle BCD=10^o
∠
BC
D
=
1
0
o
. https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.