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Michigan Mathematics Prize Competition
2010 MMPC
2010 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
Hide problems
2010 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. Let
x
1
=
0
x_1 = 0
x
1
=
0
,
x
2
=
1
/
2
x_2 = 1/2
x
2
=
1/2
and for
n
>
2
n >2
n
>
2
, let
x
n
x_n
x
n
be the average of
x
n
−
1
x_{n-1}
x
n
−
1
and
x
n
−
2
x_{n-2}
x
n
−
2
. Find a formula for
a
n
=
x
n
+
1
−
x
n
a_n = x_{n+1} - x_{n}
a
n
=
x
n
+
1
−
x
n
,
n
=
1
,
2
,
3
,
…
n = 1, 2, 3, \dots
n
=
1
,
2
,
3
,
…
. Justify your answer. p2. Given a triangle
A
B
C
ABC
A
BC
. Let
h
a
,
h
b
,
h
c
h_a, h_b, h_c
h
a
,
h
b
,
h
c
be the altitudes to its sides
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
respectively. Prove:
1
h
a
+
1
h
b
>
1
h
c
\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}
h
a
1
+
h
b
1
>
h
c
1
Is it possible to construct a triangle with altitudes
7
7
7
,
11
11
11
, and
20
20
20
? Justify your answer. p3. Does there exist a polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients such that
P
(
0
)
=
1
P(0) = 1
P
(
0
)
=
1
,
P
(
2
)
=
3
P(2) = 3
P
(
2
)
=
3
and
P
(
4
)
=
9
P(4) = 9
P
(
4
)
=
9
? Justify your answer. p4. Prove that if
cos
θ
\cos \theta
cos
θ
is rational and
n
n
n
is an integer, then
cos
n
θ
\cos n\theta
cos
n
θ
is rational. Let
α
=
1
2010
\alpha=\frac{1}{2010}
α
=
2010
1
. Is
cos
α
\cos \alpha
cos
α
rational ? Justify your answer. p5. Let function
f
(
x
)
f(x)
f
(
x
)
be defined as
f
(
x
)
=
x
2
+
b
x
+
c
f(x) = x^2 + bx + c
f
(
x
)
=
x
2
+
b
x
+
c
, where
b
,
c
b, c
b
,
c
are real numbers. (A) Evaluate
f
(
1
)
−
2
f
(
5
)
+
f
(
9
)
f(1) -2f(5) + f(9)
f
(
1
)
−
2
f
(
5
)
+
f
(
9
)
. (B) Determine all pairs
(
b
,
c
)
(b, c)
(
b
,
c
)
such that
∣
f
(
x
)
∣
≤
8
|f(x)| \le 8
∣
f
(
x
)
∣
≤
8
for all
x
x
x
in the interval
[
1
,
9
]
[1, 9]
[
1
,
9
]
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.