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1983 MMPC
1983 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
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1
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1983 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Find the largest integer which is a factor of all numbers of the form
n
(
n
+
1
)
(
n
+
2
)
n(n +1)(n + 2)
n
(
n
+
1
)
(
n
+
2
)
where
n
n
n
is any positive integer with unit digit
4
4
4
. Prove your claims. p2. Each pair of the towns
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him to call on a customer in each of the towns.Note that it is not required that he return to his starting point. https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png p3.
A
A
A
and
B
B
B
are two points on a circular race track . One runner starts at
A
A
A
running counter clockwise, and, at the same time, a second runner starts from
B
B
B
running clockwise. They meet first
100
100
100
yds from A, measured along the track. They meet a second time at
B
B
B
and the third time at
A
A
A
. Assuming constant speeds, now long is the track? p4.
A
A
A
and
B
B
B
are points on the positive
x
x
x
and positive
y
y
y
axis, respectively, and
C
C
C
is the point
(
3
,
4
)
(3,4)
(
3
,
4
)
. Prove that the perimeter of
△
A
B
C
\vartriangle ABC
△
A
BC
is greater than
10
10
10
.Suggestion: Reflect!! p5. Let
A
1
,
A
2
,
.
.
.
,
A
8
A_1,A_2,...,A_8
A
1
,
A
2
,
...
,
A
8
be a permutation of the integers
1
,
2
,
.
.
.
,
8
1,2,...,8
1
,
2
,
...
,
8
so chosen that the eight sums
9
+
A
1
9 + A_1
9
+
A
1
,
10
+
A
2
10 + A_2
10
+
A
2
,
.
.
.
...
...
,
16
+
A
8
16 + A_8
16
+
A
8
and the eight differences
9
−
A
1
9 -A_1
9
−
A
1
,
10
−
A
2
10 - A_2
10
−
A
2
,
.
.
.
...
...
,
16
−
A
8
16 - A_8
16
−
A
8
together comprise
16
16
16
different numbers. Show that the same property holds for the eight numbers in reverse order. That is, show that the
16
16
16
numbers
9
+
A
8
9 + A_8
9
+
A
8
,
10
+
A
7
10 + A_7
10
+
A
7
,
.
.
.
...
...
,
16
+
A
1
16 + A_1
16
+
A
1
and
9
−
A
8
9 -A_8
9
−
A
8
,
10
−
A
7
10 - A_7
10
−
A
7
,
.
.
.
...
...
,
16
−
A
1
16 - A_1
16
−
A
1
are also pairwise different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.