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Duke Math Meet (DMM)
2006 Duke Math Meet
2006 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
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2006 DMM Tiebreaker Round - Duke Math Meet
p1. Suppose that
a
a
a
,
b
b
b
, and
c
c
c
are positive integers such that not all of them are even,
a
<
b
a < b
a
<
b
,
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
, and
c
−
b
=
289
c - b = 289
c
−
b
=
289
. What is the smallest possible value for
c
c
c
? p2. If
a
,
b
>
1
a, b > 1
a
,
b
>
1
and
a
2
a^2
a
2
is
11
11
11
in base
b
b
b
, what is the third digit from the right of
b
2
b^2
b
2
in base
a
a
a
? p3. Find real numbers
a
,
b
a, b
a
,
b
such that
x
2
−
x
−
1
x^2 - x - 1
x
2
−
x
−
1
is a factor of
a
x
10
+
b
x
9
+
1
ax^{10} + bx^9 + 1
a
x
10
+
b
x
9
+
1
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2006 DMM Team Round - Duke Math Meet
p1. What is the smallest positive integer
x
x
x
such that
1
x
<
12011
−
12006
\frac{1}{x} <\sqrt{12011} - \sqrt{12006}
x
1
<
12011
−
12006
? p2. Two soccer players run a drill on a
100
100
100
foot by
300
300
300
foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by
100
100
100
feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of
50
50
50
feet per second, both players run at a constant speed of
30
30
30
feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld? p3. A trapezoid
A
B
C
D
ABCD
A
BC
D
has
A
B
AB
A
B
and
C
D
CD
C
D
both perpendicular to
A
D
AD
A
D
and
B
C
=
A
B
+
A
D
BC =AB + AD
BC
=
A
B
+
A
D
. If
A
B
=
26
AB = 26
A
B
=
26
, what is
C
D
2
A
D
+
C
D
\frac{CD^2}{AD+CD}
A
D
+
C
D
C
D
2
? p4. A hydrophobic, hungry, and lazy mouse is at
(
0
,
0
)
(0, 0)
(
0
,
0
)
, a piece of cheese at
(
26
,
26
)
(26, 26)
(
26
,
26
)
, and a circular lake of radius
5
2
5\sqrt2
5
2
is centered at
(
13
,
13
)
(13, 13)
(
13
,
13
)
. What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake? p5. Let
a
,
b
a, b
a
,
b
, and
c
c
c
be real numbers such that
a
+
b
+
c
=
0
a + b + c = 0
a
+
b
+
c
=
0
and
a
2
+
b
2
+
c
2
=
3
a^2 + b^2 + c^2 = 3
a
2
+
b
2
+
c
2
=
3
. If
a
5
+
b
5
+
c
5
≠
0
a^5 + b^5 + c^5\ne 0
a
5
+
b
5
+
c
5
=
0
, compute
(
a
3
+
b
3
+
c
3
)
(
a
4
+
b
4
+
c
4
)
a
5
+
b
5
+
c
5
\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}
a
5
+
b
5
+
c
5
(
a
3
+
b
3
+
c
3
)
(
a
4
+
b
4
+
c
4
)
. p6. Let
S
S
S
be the number of points with integer coordinates that lie on the line segment with endpoints
(
2
2
2
,
4
4
4
)
\left( 2^{2^2}, 4^{4^4}\right)
(
2
2
2
,
4
4
4
)
and
(
4
4
4
,
0
)
\left(4^{4^4}, 0\right)
(
4
4
4
,
0
)
. Compute
log
2
(
S
−
1
)
\log_2 (S - 1)
lo
g
2
(
S
−
1
)
. p7. For a positive integer
n
n
n
let
f
(
n
)
f(n)
f
(
n
)
be the sum of the digits of
n
n
n
. Calculate
f
(
f
(
f
(
2
2006
)
)
)
f(f(f(2^{2006})))
f
(
f
(
f
(
2
2006
)))
p8. If
a
1
,
a
2
,
a
3
,
a
4
a_1, a_2, a_3, a_4
a
1
,
a
2
,
a
3
,
a
4
are roots of
x
4
−
2006
x
3
+
11
x
+
11
=
0
x^4 - 2006x^3 + 11x + 11 = 0
x
4
−
2006
x
3
+
11
x
+
11
=
0
, fi nd
∣
a
1
3
+
a
2
3
+
a
3
3
+
a
4
3
∣
|a^3_1 + a^3_2 + a^3_3 + a^3_4|
∣
a
1
3
+
a
2
3
+
a
3
3
+
a
4
3
∣
. p9. A triangle
A
B
C
ABC
A
BC
has
M
M
M
and
N
N
N
on sides
B
C
BC
BC
and
A
C
AC
A
C
, respectively, such that
A
M
AM
A
M
and
B
N
BN
BN
intersect at
P
P
P
and the areas of triangles
A
N
P
ANP
A
NP
,
A
P
B
APB
A
PB
, and
P
M
B
PMB
PMB
are
5
5
5
,
10
10
10
, and
8
8
8
respectively. If
R
R
R
and
S
S
S
are the midpoints of
M
C
MC
MC
and
N
C
NC
NC
, respectively, compute the area of triangle
C
R
S
CRS
CRS
. p10. Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer
n
n
n
, pressing PS will cause the calculator to divide
n
n
n
by the largest power of
2
2
2
that evenly divides
n
n
n
, and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer
k
k
k
between
1
1
1
and
1023
1023
1023
, inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of
2
2
2
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2006 DMM Devil Round - Duke Math Meet
p1. The entrance fee the county fair is
64
64
64
cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? p2. At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position
A
A
A
and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position
A
A
A
, you win a giant chunk of cheese. What is the probability of winning the cheese? p3. A clown stops you and poses a riddle. How many ways can you distribute
21
21
21
identical balls into
3
3
3
different boxes, with at least
4
4
4
balls in the first box and at least
1
1
1
ball in the second box? p4. Watch out for the pig. How many sets
S
S
S
of positive integers are there such that the product of all the elements of the set is
125970
125970
125970
? p5. A good word is a word consisting of two letters
A
A
A
,
B
B
B
such that there is never a letter
B
B
B
between any two
A
A
A
's. Find the number of good words with length
8
8
8
. p6. Evaluate
2
−
2
+
2
−
.
.
.
\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}
2
−
2
+
2
−
...
without looking. p7. There is nothing wrong with being odd. Of the first
2006
2006
2006
Fibonacci numbers (
F
1
=
1
F_1 = 1
F
1
=
1
,
F
2
=
1
F_2 = 1
F
2
=
1
), how many of them are even? p8. Let
f
f
f
be a function satisfying
f
(
x
)
+
2
f
(
27
−
x
)
=
x
f (x) + 2f (27- x) = x
f
(
x
)
+
2
f
(
27
−
x
)
=
x
. Find
f
(
11
)
f (11)
f
(
11
)
. p9. Let
A
A
A
,
B
B
B
,
C
C
C
denote digits in decimal representation. Given that
A
A
A
is prime and
A
−
B
=
4
A -B = 4
A
−
B
=
4
, nd
(
A
,
B
,
C
)
(A,B,C)
(
A
,
B
,
C
)
such that
A
A
A
B
B
B
C
AAABBBC
AAA
BBBC
is a prime. p10. Given
x
2
+
y
2
x
2
−
y
2
+
x
2
−
y
2
x
2
+
y
2
=
k
\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k
x
2
−
y
2
x
2
+
y
2
+
x
2
+
y
2
x
2
−
y
2
=
k
, fi nd
x
8
+
y
8
x
8
−
y
8
\frac{x^8+y^8}{x^8-y^8}
x
8
−
y
8
x
8
+
y
8
in term of
k
k
k
. p11. Let
a
i
∈
{
−
1
,
0
,
1
}
a_i \in \{-1, 0, 1\}
a
i
∈
{
−
1
,
0
,
1
}
for each
i
=
1
,
2
,
3
,
.
.
.
,
2007
i = 1, 2, 3, ..., 2007
i
=
1
,
2
,
3
,
...
,
2007
. Find the least possible value for
∑
i
=
1
2006
∑
j
=
i
+
1
2007
a
i
a
j
\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j
∑
i
=
1
2006
∑
j
=
i
+
1
2007
a
i
a
j
. p12. Find all integer solutions
x
x
x
to
x
2
+
615
=
2
n
x^2 + 615 = 2^n
x
2
+
615
=
2
n
for any integer
n
≥
1
n \ge 1
n
≥
1
. p13. Suppose a parabola
y
=
x
2
−
a
x
−
1
y = x^2 - ax - 1
y
=
x
2
−
a
x
−
1
intersects the coordinate axes at three points
A
A
A
,
B
B
B
, and
C
C
C
. The circumcircle of the triangle
A
B
C
ABC
A
BC
intersects the
y
y
y
- axis again at point
D
=
(
0
,
t
)
D = (0, t)
D
=
(
0
,
t
)
. Find the value of
t
t
t
. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.