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Michigan Mathematics Prize Competition
1992 MMPC
1992 MMPC
Part of
Michigan Mathematics Prize Competition
Subcontests
(1)
1
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1992 MMPC , Part 2 = Michigan Mathematics Prize Competition
p1. The English alphabet consists of
21
21
21
consonants and
5
5
5
vowels. (We count
y
y
y
as a consonant.) (a) Suppose that all the letters are listed in an arbitrary order. Prove that there must be
4
4
4
consecutive consonants. (b) Give a list to show that there need not be
5
5
5
consecutive consonants. (c) Suppose that all the letters are arranged in a circle. Prove that there must be
5
5
5
consecutive consonants. p2. From the set
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1,2,3,... , n\}
{
1
,
2
,
3
,
...
,
n
}
,
k
k
k
distinct integers are selected at random and arranged in numerical order (lowest to highest). Let
P
(
i
,
r
,
k
,
n
)
P(i, r, k, n)
P
(
i
,
r
,
k
,
n
)
denote the probability that integer
i
i
i
is in position
r
r
r
. For example, observe that
P
(
1
,
2
,
k
,
n
)
=
0
P(1, 2, k, n) = 0
P
(
1
,
2
,
k
,
n
)
=
0
. (a) Compute
P
(
2
,
1
,
6
,
10
)
P(2, 1,6,10)
P
(
2
,
1
,
6
,
10
)
. (b) Find a general formula for
P
(
i
,
r
,
k
,
n
)
P(i, r, k, n)
P
(
i
,
r
,
k
,
n
)
. p3. (a) Write down a fourth degree polynomial
P
(
x
)
P(x)
P
(
x
)
such that
P
(
1
)
=
P
(
−
1
)
P(1) = P(-1)
P
(
1
)
=
P
(
−
1
)
but
P
(
2
)
≠
P
(
−
2
)
P(2) \ne P(-2)
P
(
2
)
=
P
(
−
2
)
(b) Write down a fifth degree polynomial
Q
(
x
)
Q(x)
Q
(
x
)
such that
Q
(
1
)
=
Q
(
−
1
)
Q(1) = Q(-1)
Q
(
1
)
=
Q
(
−
1
)
and
Q
(
2
)
=
Q
(
−
2
)
Q(2) = Q(-2)
Q
(
2
)
=
Q
(
−
2
)
but
Q
(
3
)
≠
Q
(
−
3
)
Q(3) \ne Q(-3)
Q
(
3
)
=
Q
(
−
3
)
. (c) Prove that, if a sixth degree polynomial
R
(
x
)
R(x)
R
(
x
)
satisfies
R
(
1
)
=
R
(
−
1
)
R(1) = R(-1)
R
(
1
)
=
R
(
−
1
)
,
R
(
2
)
=
R
(
−
2
)
R(2) = R(-2)
R
(
2
)
=
R
(
−
2
)
, and
R
(
3
)
=
R
(
−
3
)
R(3) = R(-3)
R
(
3
)
=
R
(
−
3
)
, then
R
(
x
)
=
R
(
−
x
)
R(x) = R(-x)
R
(
x
)
=
R
(
−
x
)
for all
x
x
x
. p4. Given five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number
N
N
N
of distinct values among these sums. (a) Give an example of five numbers yielding the smallest possible value of
N
N
N
. What is this value? (b) Give an example of five numbers yielding the largest possible value of
N
N
N
. What is this value? (c) Prove that the values of
N
N
N
you obtained in (a) and (b) are the smallest and largest possible ones. p5. Let
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
be a triangle which is not a right triangle. Prove that there exist circles
C
1
C_1
C
1
,
C
2
C_2
C
2
, and
C
3
C_3
C
3
such that
C
2
C_2
C
2
is tangent to
C
3
C_3
C
3
at
A
1
A_1
A
1
,
C
3
C_3
C
3
is tangent to
C
1
C_1
C
1
at
A
2
A_2
A
2
, and
C
1
C_1
C
1
is tangent to
C
2
C_2
C
2
at
A
3
A_3
A
3
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.