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Michigan Mathematics Prize Competition
1971 MMPC
1971 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1971 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Prove that there is no interger
n
n
n
such that
n
2
+
1
n^2 +1
n
2
+
1
is divisible by
7
7
7
. p2. Find all solutions of the system
x
2
−
y
z
=
1
x^2-yz=1
x
2
−
yz
=
1
y
2
−
x
z
=
2
y^2-xz=2
y
2
−
x
z
=
2
z
2
−
x
y
=
3
z^2-xy=3
z
2
−
x
y
=
3
p3. A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number,
n
n
n
, of points in the plane with the property that every three of them form the vertices of a triangle with long legs? Prove all assertions. p4. Prove that the area of a quadrilateral of sides
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
which can be inscribed in a circle and circumscribed about another circle is given by
A
=
a
b
c
d
A=\sqrt{abcd}
A
=
ab
c
d
p5. Prove that all of the squares of side length
1
2
,
1
3
,
1
4
,
1
5
,
1
6
,
.
.
.
,
1
n
,
.
.
.
\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...
2
1
,
3
1
,
4
1
,
5
1
,
6
1
,
...
,
n
1
,
...
can fit inside a square of side length
1
1
1
without overlap. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.