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Michigan Mathematics Prize Competition
1974 MMPC
1974 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
Hide problems
1974 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Let
S
S
S
be the sum of the
99
99
99
terms:
(
1
+
2
)
−
1
,
(
2
+
3
)
−
1
,
(
3
+
4
)
−
1
,
.
.
.
,
(
99
+
100
)
−
1
.
(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.
(
1
+
2
)
−
1
,
(
2
+
3
)
−
1
,
(
3
+
4
)
−
1
,
...
,
(
99
+
100
)
−
1
.
Prove that
S
S
S
is an integer. p2. Determine all pairs of positive integers
x
x
x
and
y
y
y
for which
N
=
x
4
+
4
y
4
N=x^4+4y^4
N
=
x
4
+
4
y
4
is a prime. (Your work should indicate why no other solutions are possible.) p3. Let
w
,
x
,
y
,
z
w,x,y,z
w
,
x
,
y
,
z
be arbitrary positive real numbers. Prove each inequality:(a)
x
y
≤
(
x
+
y
2
)
2
xy \le \left(\frac{x+y}{2}\right)^2
x
y
≤
(
2
x
+
y
)
2
(b)
w
x
y
z
≤
(
w
+
x
+
y
+
z
4
)
4
wxyz \le \left(\frac{w+x+y+z}{4}\right)^4
w
x
yz
≤
(
4
w
+
x
+
y
+
z
)
4
(c)
x
y
z
≤
(
x
+
y
+
z
3
)
3
xyz \le \left(\frac{x+y+z}{3}\right)^3
x
yz
≤
(
3
x
+
y
+
z
)
3
p4. Twelve points
P
1
P_1
P
1
,
P
2
P_2
P
2
,
.
.
.
...
...
,
P
12
P_{12}
P
12
are equally spaaed on a circle, as shown. Prove: that the chords
P
1
P
9
‾
\overline{P_1P_9}
P
1
P
9
,
P
4
P
12
‾
\overline{P_4P_{12}}
P
4
P
12
and
P
2
P
11
‾
\overline{P_2P_{11}}
P
2
P
11
have a point in common. https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png p5. Two very busy men,
A
A
A
and
B
B
B
, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than
12
:
15
12:15
12
:
15
p.m. If necessary,
A
A
A
will wait
6
6
6
minutes for
B
B
B
to arrive, while
B
B
B
will wait
9
9
9
minutes for
A
A
A
to arrive but neither can stay past
12
:
15
12:15
12
:
15
p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.