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Michigan Mathematics Prize Competition
1963 MMPC
1963 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
Hide problems
1963 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Suppose
x
≠
1
x \ne 1
x
=
1
or
10
10
10
and logarithms are computed to the base
10
10
10
. Define
y
=
1
0
1
1
−
log
x
y= 10^{\frac{1}{1-\log x}}
y
=
1
0
1
−
l
o
g
x
1
and
z
=
1
1
−
log
y
z = ^{\frac{1}{1-\log y}}
z
=
1
−
l
o
g
y
1
. Prove that
x
=
1
0
1
1
−
log
z
x= 10^{\frac{1}{1-\log z}}
x
=
1
0
1
−
l
o
g
z
1
p2. If
n
n
n
is an odd number and
x
1
,
x
2
,
x
3
,
.
.
.
,
x
n
x_1, x_2, x_3,..., x_n
x
1
,
x
2
,
x
3
,
...
,
x
n
is an arbitrary arrangement of the integers
1
,
2
,
3
,
.
.
.
,
n
1, 2,3,..., n
1
,
2
,
3
,
...
,
n
, prove that the product
(
x
1
−
1
)
(
x
2
−
2
)
(
x
3
−
3
)
.
.
.
(
x
n
−
n
)
(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)
(
x
1
−
1
)
(
x
2
−
2
)
(
x
3
−
3
)
...
(
x
n
−
n
)
is an even number (possibly negative or zero). p3. Prove that
1
⋅
3
⋅
5
⋅
⋅
⋅
(
2
n
−
1
)
2
⋅
4
⋅
6
⋅
⋅
⋅
(
2
n
<
1
2
n
+
1
\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}
2
⋅
4
⋅
6
⋅⋅⋅
(
2
n
1
⋅
3
⋅
5
⋅⋅⋅
(
2
n
−
1
)
<
2
n
+
1
1
for all integers
n
=
1
,
2
,
3
,
.
.
.
n = 1,2,3,...
n
=
1
,
2
,
3
,
...
p4. Prove that if three angles of a convex polygon are each
6
0
o
60^o
6
0
o
, then the polygon must be an equilateral triangle. p5. Find all solutions, real and complex, of
4
(
x
2
+
1
x
2
)
−
4
(
x
+
1
x
)
−
7
=
0
4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0
4
(
x
2
+
x
2
1
)
−
4
(
x
+
x
1
)
−
7
=
0
p6. A man is
3
8
\frac38
8
3
of the way across a narrow railroad bridge when he hears a train approaching at
60
60
60
miles per hour. No matter which way he runs he can just escape being hit by the train. How fast can he run? Prove your assertion.PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.