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2008 MMPC
2008 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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2008 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Compute
(
1
10
)
1
2
(
1
1
0
2
)
1
2
4
(
1
1
0
3
)
1
2
3
.
.
.
\left(\frac{1}{10}\right)^{\frac12}\left(\frac{1}{10^2}\right)^{\frac{1}{2^4}}\left(\frac{1}{10^3}\right)^{\frac{1}{2^3}} ...
(
10
1
)
2
1
(
1
0
2
1
)
2
4
1
(
1
0
3
1
)
2
3
1
...
p2. Consider the sequence
1
,
2
,
2
,
3
,
3
,
3
,
4
,
4
,
4
,
4
,
.
.
.
,
1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...,
1
,
2
,
2
,
3
,
3
,
3
,
4
,
4
,
4
,
4
,
...
,
where the positive integer
m
m
m
appears
m
m
m
times. Let
d
(
n
)
d(n)
d
(
n
)
denote the
n
n
n
th element of this sequence starting with
n
=
1
n = 1
n
=
1
. Find a closed-form formula for
d
(
n
)
d(n)
d
(
n
)
. p3. Let
0
<
θ
<
π
2
0 < \theta < \frac{\pi}{2}
0
<
θ
<
2
π
, prove that
(
sin
2
θ
2
+
2
cos
2
θ
)
1
4
+
(
cos
2
θ
2
+
2
sin
2
θ
)
1
4
≥
(
68
)
1
4
\left( \frac{\sin^2 \theta}{2}+\frac{2}{\cos^2 \theta} \right)^{\frac14}+ \left( \frac{\cos^2 \theta}{2}+\frac{2}{\sin^2 \theta} \right)^{\frac14} \ge (68)^{\frac14}
(
2
sin
2
θ
+
cos
2
θ
2
)
4
1
+
(
2
cos
2
θ
+
sin
2
θ
2
)
4
1
≥
(
68
)
4
1
and determine the value of \theta when the inequality holds as equality. p4. In
△
A
B
C
\vartriangle ABC
△
A
BC
, parallel lines to
A
B
AB
A
B
and
A
C
AC
A
C
are drawn from a point
Q
Q
Q
lying on side
B
C
BC
BC
. If
a
a
a
is used to represent the ratio of the area of parallelogram
A
D
Q
E
ADQE
A
D
QE
to the area of the triangle
△
A
B
C
\vartriangle ABC
△
A
BC
, (i) find the maximum value of
a
a
a
. (ii) find the ratio
B
Q
Q
C
\frac{BQ}{QC}
QC
BQ
when
a
=
24
49
.
a =\frac{24}{49}.
a
=
49
24
.
https://cdn.artofproblemsolving.com/attachments/5/8/eaa58df0d55e6e648855425e581a6ba0ad3ea6.pngp5. Prove the following inequality
1
2009
<
1
2
⋅
3
4
⋅
5
6
⋅
7
8
.
.
.
2007
2008
<
1
40
\frac{1}{2009} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8}...\frac{2007}{2008}<\frac{1}{40}
2009
1
<
2
1
⋅
4
3
⋅
6
5
⋅
8
7
...
2008
2007
<
40
1
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.Thanks to gauss202 for sending the problems.