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Michigan Mathematics Prize Competition
1958 MMPC
1958 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1958 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Show that
9
x
+
5
y
9x + 5y
9
x
+
5
y
is a multiple of
17
17
17
whenever
2
x
+
3
y
2x + 3y
2
x
+
3
y
is a multiple of
17
17
17
. p2. Express the five distinct fifth roots of
1
1
1
in terms of radicals. p3. Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. p4. Find the volume of a sphere which circumscribes a regular tetrahedron of edge
a
a
a
. p5. For any integer
n
n
n
greater than
1
1
1
, show that
n
2
−
2
n
+
1
n^2-2n + 1
n
2
−
2
n
+
1
is a factor at
n
n
−
1
−
1
n^{n-1}-1
n
n
−
1
−
1
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.