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Michigan Mathematics Prize Competition
1972 MMPC
1972 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1972 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is
180
180
180
degrees. Prove that all faces of the tetrahedron are congruent triangles. https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png p2. The digital sum
D
(
n
)
D(n)
D
(
n
)
of a positive integer
n
n
n
is defined recursively by:
D
(
n
)
=
n
D(n) = n
D
(
n
)
=
n
if
1
≤
n
≤
9
1 \le n \le 9
1
≤
n
≤
9
D
(
n
)
=
D
(
a
0
+
a
1
+
.
.
.
+
a
m
)
D(n) = D(a_0 + a_1 + ... + a_m)
D
(
n
)
=
D
(
a
0
+
a
1
+
...
+
a
m
)
if
n
>
9
n>9
n
>
9
where
a
0
,
a
1
,
.
.
,
a
m
a_0 , a_1 ,..,a_m
a
0
,
a
1
,
..
,
a
m
are all the digits of
n
n
n
expressed in base ten. (For example,
D
(
959
)
=
D
(
26
)
=
D
(
8
)
=
8
D(959) = D(26) = D(8) = 8
D
(
959
)
=
D
(
26
)
=
D
(
8
)
=
8
.) Prove that
D
(
n
×
1234
)
=
D
(
n
)
D(n \times 1234)= D(n)
D
(
n
×
1234
)
=
D
(
n
)
fcr all positive integers
n
n
n
. p3. A right triangle has area
A
A
A
and perimeter
P
P
P
. Find the largest possible value for the positive constant
k
k
k
such that for every such triangle,
P
2
≥
k
A
P^2 \ge kA
P
2
≥
k
A
. p4. In the accompanying diagram,
A
B
‾
\overline{AB}
A
B
is tangent at
A
A
A
to a circle of radius
1
1
1
centered at
O
O
O
. The segment
A
P
‾
\overline{AP}
A
P
is equal in length to the arc
A
B
AB
A
B
. Let
C
C
C
be the point of intersection of the lines
A
O
AO
A
O
and
P
B
PB
PB
. Determine the length of segment
A
C
‾
\overline{AC}
A
C
in terms of
a
a
a
, where
a
a
a
is the measure of
∠
A
O
B
\angle AOB
∠
A
OB
in radians. https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png p5. Let
a
1
=
a
>
0
a_1 = a > 0
a
1
=
a
>
0
and
a
2
=
b
>
a
a_2 = b >a
a
2
=
b
>
a
. Consider the sequence
{
a
1
,
a
2
,
a
3
,
.
.
.
}
\{a_1,a_2,a_3,...\}
{
a
1
,
a
2
,
a
3
,
...
}
of positive numbers defined by:
a
3
=
a
1
a
2
a_3=\sqrt{a_1a_2}
a
3
=
a
1
a
2
,
a
4
=
a
2
a
3
a_4=\sqrt{a_2a_3}
a
4
=
a
2
a
3
,
.
.
.
...
...
and in general,
a
n
=
a
n
−
2
a
n
−
1
a_n=\sqrt{a_{n-2}a_{n-1}}
a
n
=
a
n
−
2
a
n
−
1
, for
n
≥
3
n\ge 3
n
≥
3
. Develop a formula
a
n
a_n
a
n
expressing in terms of
a
a
a
,
b
b
b
and
n
n
n
, and determine
lim
n
→
∞
a
n
\lim_{n \to \infty} a_n
lim
n
→
∞
a
n
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.