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Duke Math Meet (DMM)
2005 Duke Math Meet
2005 Duke Math Meet
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Duke Math Meet (DMM)
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2005 DMM Team Round - Duke Math Meet
p1. Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation
x
17
−
17
x
+
17
=
0
x^{17} - 17x + 17 = 0
x
17
−
17
x
+
17
=
0
. p2. Four identical spherical cows, each of radius
17
17
17
meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? p3. If
a
n
a_n
a
n
is the last digit of
∑
i
=
1
n
i
\sum^{n}_{i=1} i
∑
i
=
1
n
i
, what would the value of
∑
i
=
1
1000
a
i
\sum^{1000}_{i=1}a_i
∑
i
=
1
1000
a
i
be? p4. If there are
15
15
15
teams to play in a tournament,
2
2
2
teams per game, in how many ways can the tournament be organized if each team is to participate in exactly
5
5
5
games against dierent opponents? p5. For
n
=
20
n = 20
n
=
20
and
k
=
6
k = 6
k
=
6
, calculate
2
k
(
n
0
)
(
n
k
)
−
2
k
−
1
(
n
1
)
(
n
−
1
k
−
1
)
+
2
k
−
2
(
n
2
)
(
n
−
2
k
−
2
)
+
.
.
.
+
(
−
1
)
k
(
n
k
)
(
n
−
k
0
)
2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}
2
k
(
0
n
)
(
k
n
)
−
2
k
−
1
(
1
n
)
(
k
−
1
n
−
1
)
+
2
k
−
2
(
2
n
)
(
k
−
2
n
−
2
)
+
...
+
(
−
1
)
k
(
k
n
)
(
0
n
−
k
)
where
(
n
k
)
{n \choose k}
(
k
n
)
is the number of ways to choose
k
k
k
things from a set of
n
n
n
. p6. Given a function
f
(
x
)
=
a
x
2
+
b
f(x) = ax^2 + b
f
(
x
)
=
a
x
2
+
b
, with a
,
b
, b
,
b
real numbers such that
f
(
f
(
f
(
x
)
)
)
=
−
128
x
8
+
128
3
x
6
−
16
22
x
2
+
23
102
f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}
f
(
f
(
f
(
x
)))
=
−
128
x
8
+
3
128
x
6
−
22
16
x
2
+
102
23
, find
b
a
b^a
b
a
. p7. Simplify the following fraction
(
2
3
−
1
)
(
3
3
−
1
)
.
.
.
(
10
0
3
−
1
)
(
2
3
+
1
)
(
3
3
+
1
)
.
.
.
(
10
0
3
+
1
)
\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}
(
2
3
+
1
)
(
3
3
+
1
)
...
(
10
0
3
+
1
)
(
2
3
−
1
)
(
3
3
−
1
)
...
(
10
0
3
−
1
)
p8. Simplify the following expression
3
+
5
+
3
−
5
3
−
8
−
4
8
−
2
15
\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}
3
−
8
3
+
5
+
3
−
5
−
8
−
2
15
4
p9. Suppose that
p
(
x
)
p(x)
p
(
x
)
is a polynomial of degree
100
100
100
such that
p
(
k
)
=
k
2
k
−
1
p(k) = k2^{k-1}
p
(
k
)
=
k
2
k
−
1
,
k
=
1
,
2
,
3
,
.
.
.
,
100
k =1, 2, 3 ,... , 100
k
=
1
,
2
,
3
,
...
,
100
. What is the value of
p
(
101
)
p(101)
p
(
101
)
? p10. Find all
17
17
17
real solutions
(
w
,
x
,
y
,
z
)
(w, x, y, z)
(
w
,
x
,
y
,
z
)
to the following system of equalities:
2
w
+
w
2
x
=
x
2w + w^2x = x
2
w
+
w
2
x
=
x
2
x
+
x
2
y
=
y
2x + x^2y=y
2
x
+
x
2
y
=
y
2
y
+
y
2
z
=
z
2y + y^2z=z
2
y
+
y
2
z
=
z
−
2
z
+
z
2
w
=
w
-2z+z^2w=w
−
2
z
+
z
2
w
=
w
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2005 DMM Devil Round - Duke Math Meet
p1. Let
a
a
a
and
b
b
b
be complex numbers such that
a
3
+
b
3
=
−
17
a^3 + b^3 = -17
a
3
+
b
3
=
−
17
and
a
+
b
=
1
a + b = 1
a
+
b
=
1
. What is the value of
a
b
ab
ab
? p2. Let
A
E
F
B
AEFB
A
EFB
be a right trapezoid, with
∠
A
E
F
=
∠
E
A
B
=
9
0
o
\angle AEF = \angle EAB = 90^o
∠
A
EF
=
∠
E
A
B
=
9
0
o
. The two diagonals
E
B
EB
EB
and
A
F
AF
A
F
intersect at point
D
D
D
, and
C
C
C
is a point on
A
E
AE
A
E
such that
A
E
⊥
D
C
AE \perp DC
A
E
⊥
D
C
. If
A
B
=
8
AB = 8
A
B
=
8
and
E
F
=
17
EF = 17
EF
=
17
, what is the lenght of
C
D
CD
C
D
? p3. How many three-digit numbers
a
b
c
abc
ab
c
(where each of
a
a
a
,
b
b
b
, and
c
c
c
represents a single digit,
a
≠
0
a \ne 0
a
=
0
) are there such that the six-digit number
a
b
c
a
b
c
abcabc
ab
c
ab
c
is divisible by
2
2
2
,
3
3
3
,
5
5
5
,
7
7
7
,
11
11
11
, or
13
13
13
? p4. Let
S
S
S
be the sum of all numbers of the form
1
n
\frac{1}{n}
n
1
where
n
n
n
is a postive integer and
1
n
\frac{1}{n}
n
1
terminales in base
b
b
b
, a positive integer. If
S
S
S
is
15
8
\frac{15}{8}
8
15
, what is the smallest such
b
b
b
? p5. Sysyphus is having an birthday party and he has a square cake that is to be cut into
25
25
25
square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram? p6. Given
(
9
x
2
−
y
2
)
(
9
x
2
+
6
x
y
+
y
2
)
=
16
(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16
(
9
x
2
−
y
2
)
(
9
x
2
+
6
x
y
+
y
2
)
=
16
and
3
x
+
y
=
2
3x + y = 2
3
x
+
y
=
2
. Find
x
y
x^y
x
y
. p7. What is the prime factorization of the smallest integer
N
N
N
such that
N
2
\frac{N}{2}
2
N
is a perfect square,
N
3
\frac{N}{3}
3
N
is a perfect cube,
N
5
\frac{N}{5}
5
N
is a perfect fifth power? p8. What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts? p9. How many ordered triples of integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
are there such that
0
≤
x
,
y
,
z
≤
100
0 \le x, y, z \le 100
0
≤
x
,
y
,
z
≤
100
and
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
≥
(
x
+
y
−
2
z
)
+
(
y
+
z
−
2
x
)
2
+
(
z
+
x
−
2
y
)
2
.
(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
≥
(
x
+
y
−
2
z
)
+
(
y
+
z
−
2
x
)
2
+
(
z
+
x
−
2
y
)
2
.
p10. Find all real solutions to
(
2
x
−
4
)
2
+
(
4
x
−
2
)
3
=
(
4
x
+
2
x
−
6
)
3
(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3
(
2
x
−
4
)
2
+
(
4
x
−
2
)
3
=
(
4
x
+
2
x
−
6
)
3
. p11. Let
f
f
f
be a function that takes integers to integers that also has
f
(
x
)
=
{
x
−
5
i
f
x
≥
50
f
(
f
(
x
+
12
)
)
i
f
x
<
50
f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}
f
(
x
)
=
{
x
−
5
i
f
x
≥
50
f
(
f
(
x
+
12
))
i
f
x
<
50
Evaluate
f
(
2
)
+
f
(
39
)
+
f
(
58
)
.
f (2) + f (39) + f (58).
f
(
2
)
+
f
(
39
)
+
f
(
58
)
.
p12. If two real numbers are chosen at random (i.e. uniform distribution) from the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
, what is the probability that theit difference will be less than
3
5
\frac35
5
3
? p13. Let
a
a
a
,
b
b
b
, and
c
c
c
be positive integers, not all even, such that
a
<
b
a < b
a
<
b
,
b
=
c
−
2
b = c - 2
b
=
c
−
2
, and
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
. What is the smallest possible value for
c
c
c
? p14. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral whose diagonals intersect at
O
O
O
. If
B
O
=
8
BO = 8
BO
=
8
,
O
D
=
8
OD = 8
O
D
=
8
,
A
O
=
16
AO = 16
A
O
=
16
,
O
C
=
4
OC = 4
OC
=
4
, and
A
B
=
16
AB = 16
A
B
=
16
, then find
A
D
AD
A
D
. p15. Let
P
0
P_0
P
0
be a regular icosahedron with an edge length of
17
17
17
units. For each nonnegative integer
n
n
n
, recursively construct
P
n
+
1
P_{n+1}
P
n
+
1
from Pn by performing the following procedure on each face of
P
n
P_n
P
n
: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of
P
n
P_n
P
n
. Express the number of square units in the surface area of
P
17
P_{17}
P
17
in the form
u
v
⋅
w
x
y
z
\frac{u^v\cdot w \sqrt{x}}{y^z}
y
z
u
v
⋅
w
x
, where
u
,
v
,
w
,
x
,
y
u, v, w, x, y
u
,
v
,
w
,
x
,
y
, and
z
z
z
are integers, all greater than or equal to
2
2
2
, that satisfy the following conditions: the only perfect square that evenly divides
x
x
x
is
1
1
1
, the GCD of
u
u
u
and y is
1
1
1
, and neither
u
u
u
nor
y
y
y
divides
w
w
w
. Answers written in any other form will not be considered correct! PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.