MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Duke Math Meet (DMM)
2019 Duke Math Meet
2019 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
(1)
3
Hide problems
2019 DMM Tiebreaker Round - Duke Math Meet
p1. Let
a
(
1
)
,
a
(
2
)
,
.
.
.
,
a
(
n
)
,
.
.
.
a(1), a(2), ..., a(n),...
a
(
1
)
,
a
(
2
)
,
...
,
a
(
n
)
,
...
be an increasing sequence of positive integers satisfying
a
(
a
(
n
)
)
=
3
n
a(a(n)) = 3n
a
(
a
(
n
))
=
3
n
for every positive integer
n
n
n
. Compute
a
(
2019
)
a(2019)
a
(
2019
)
. p2. Consider the function
f
(
12
x
−
7
)
=
18
x
3
−
5
x
+
1
f(12x - 7) = 18x^3 - 5x + 1
f
(
12
x
−
7
)
=
18
x
3
−
5
x
+
1
. Then,
f
(
x
)
f(x)
f
(
x
)
can be expressed as
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x) = ax^3 + bx^2 + cx + d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
, for some real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
. Find the value of
(
a
+
c
)
(
b
+
d
)
(a + c)(b + d)
(
a
+
c
)
(
b
+
d
)
. p3. Let
a
,
b
a, b
a
,
b
be real numbers such that
5
+
2
6
=
a
+
b
\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}
5
+
2
6
=
a
+
b
. Find the largest value of the quantity
X
=
1
a
+
1
b
+
1
a
+
.
.
.
X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}
X
=
a
+
b
+
a
+
...
1
1
1
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2019 DMM Team Round - Duke Math Meet
p1. Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? p2. Suppose we wrote the integers between
0001
0001
0001
and
2019
2019
2019
on a blackboard as such:
000100020003
⋅
⋅
⋅
20182019.
000100020003 · · · 20182019.
000100020003
⋅⋅⋅
20182019.
How many
0
0
0
’s did we write? p3. Duke’s basketball team has made
x
x
x
three-pointers,
y
y
y
two-pointers, and
z
z
z
one-point free throws, where
x
,
y
,
z
x, y, z
x
,
y
,
z
are whole numbers. Given that
3
∣
x
3|x
3∣
x
,
5
∣
y
5|y
5∣
y
, and
7
∣
z
7|z
7∣
z
, find the greatest number of points that Duke’s basketball team could not have scored. p4. Find the minimum value of
x
2
+
2
x
y
+
3
y
2
+
4
x
+
8
y
+
12
x^2 + 2xy + 3y^2 + 4x + 8y + 12
x
2
+
2
x
y
+
3
y
2
+
4
x
+
8
y
+
12
, given that
x
x
x
and
y
y
y
are real numbers. Note: calculus is not required to solve this problem. p5. Circles
C
1
,
C
2
C_1, C_2
C
1
,
C
2
have radii
1
,
2
1, 2
1
,
2
and are centered at
O
1
,
O
2
O_1, O_2
O
1
,
O
2
, respectively. They intersect at points
A
A
A
and
B
B
B
, and convex quadrilateral
O
1
A
O
2
B
O_1AO_2B
O
1
A
O
2
B
is cyclic. Find the length of
A
B
AB
A
B
. Express your answer as
x
/
y
x/\sqrt{y}
x
/
y
, where
x
,
y
x, y
x
,
y
are integers and
y
y
y
is square-free. p6. An infinite geometric sequence
{
a
n
}
\{a_n\}
{
a
n
}
has sum
∑
n
=
0
∞
a
n
=
3
\sum_{n=0}^{\infty} a_n = 3
∑
n
=
0
∞
a
n
=
3
. Compute the maximum possible value of the sum
∑
n
=
0
∞
a
3
n
\sum_{n=0}^{\infty} a_{3n}
∑
n
=
0
∞
a
3
n
. p7. Let there be a sequence of numbers
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3,...
x
1
,
x
2
,
x
3
,
...
such that for all
i
i
i
,
x
i
=
49
7
i
1010
+
49
.
x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.
x
i
=
7
1010
i
+
49
49
.
Find the largest value of
n
n
n
such that
⌊
∑
i
=
1
n
x
i
⌋
≤
2019.
\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.
⌊
i
=
1
∑
n
x
i
⌋
≤
2019.
p8. Let
X
X
X
be a
9
9
9
-digit integer that includes all the digits
1
1
1
through
9
9
9
exactly once, such that any
2
2
2
-digit number formed from adjacent digits of
X
X
X
is divisible by
7
7
7
or
13
13
13
. Find all possible values of
X
X
X
. p9. Two
2025
2025
2025
-digit numbers,
428
99... 99
⏟
2019
9’s
571
428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571
428
2019
9’s
99... 99
571
and
571
99... 99
⏟
2019
9’s
428
571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428
571
2019
9’s
99... 99
428
, form the legs of a right triangle. Find the sum of the digits in the hypotenuse. p10. Suppose that the side lengths of
△
A
B
C
\vartriangle ABC
△
A
BC
are positive integers and the perimeter of the triangle is
35
35
35
. Let
G
G
G
the centroid and
I
I
I
be the incenter of the triangle. Given that
∠
G
I
C
=
9
0
o
\angle GIC = 90^o
∠
G
I
C
=
9
0
o
, what is the length of
A
B
AB
A
B
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2019 DMM Individual Round - Duke Math Meet
p1. Compute the value of
N
N
N
, where
N
=
81
8
3
−
6
⋅
81
8
2
⋅
209
+
12
⋅
818
⋅
20
9
2
−
8
⋅
20
9
3
N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3
N
=
81
8
3
−
6
⋅
81
8
2
⋅
209
+
12
⋅
818
⋅
20
9
2
−
8
⋅
20
9
3
p2. Suppose
x
≤
2019
x \le 2019
x
≤
2019
is a positive integer that is divisible by
2
2
2
and
5
5
5
, but not
3
3
3
. If
7
7
7
is one of the digits in
x
x
x
, how many possible values of
x
x
x
are there? p3. Find all non-negative integer solutions
(
a
,
b
)
(a,b)
(
a
,
b
)
to the equation
b
2
+
b
+
1
=
a
2
.
b^2 + b + 1 = a^2.
b
2
+
b
+
1
=
a
2
.
p4. Compute the remainder when
∑
n
=
1
2019
n
4
\sum^{2019}_{n=1} n^4
∑
n
=
1
2019
n
4
is divided by
53
53
53
. p5. Let
A
B
C
ABC
A
BC
be an equilateral triangle and
C
D
E
F
CDEF
C
D
EF
a square such that
E
E
E
lies on segment
A
B
AB
A
B
and
F
F
F
on segment
B
C
BC
BC
. If the perimeter of the square is equal to
4
4
4
, what is the area of triangle
A
B
C
ABC
A
BC
? https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.pngp6.
S
=
4
1
×
2
×
3
+
5
2
×
3
×
4
+
6
3
×
4
×
5
+
.
.
.
+
101
98
×
99
×
100
S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}
S
=
1
×
2
×
3
4
+
2
×
3
×
4
5
+
3
×
4
×
5
6
+
...
+
98
×
99
×
100
101
Let
T
=
5
4
−
S
T = \frac54 - S
T
=
4
5
−
S
. If
T
=
m
n
T = \frac{m}{n}
T
=
n
m
, where
m
m
m
and
n
n
n
are relatively prime integers, find the value of
m
+
n
m + n
m
+
n
. p7. Find the sum of
∑
i
=
0
2019
2
i
2
i
+
2
2019
−
i
\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}
i
=
0
∑
2019
2
i
+
2
2019
−
i
2
i
p8. Let
A
A
A
and
B
B
B
be two points in the Cartesian plane such that
A
A
A
lies on the line
y
=
12
y = 12
y
=
12
, and
B
B
B
lies on the line
y
=
3
y = 3
y
=
3
. Let
C
1
C_1
C
1
,
C
2
C_2
C
2
be two distinct circles that intersect both
A
A
A
and
B
B
B
and are tangent to the
x
x
x
-axis at
P
P
P
and
Q
Q
Q
, respectively. If
P
Q
=
420
PQ = 420
PQ
=
420
, determine the length of
A
B
AB
A
B
. p9. Zion has an average
2
2
2
out of
3
3
3
hit rate for
2
2
2
-pointers and
1
1
1
out of
3
3
3
hit rate for
3
3
3
-pointers. In a recent basketball match, Zion scored
18
18
18
points without missing a shot, and all the points came from
2
2
2
or
3
3
3
-pointers. What is the probability that all his shots were
3
3
3
-pointers? p10. Let
S
=
{
1
,
2
,
3
,
.
.
.
,
2019
}
S = \{1,2, 3,..., 2019\}
S
=
{
1
,
2
,
3
,
...
,
2019
}
. Find the number of non-constant functions
f
:
S
→
S
f : S \to S
f
:
S
→
S
such that
f
(
k
)
=
f
(
f
(
k
+
1
)
)
≤
f
(
k
+
1
)
f
o
r
a
l
l
1
≤
k
≤
2018.
f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.
f
(
k
)
=
f
(
f
(
k
+
1
))
≤
f
(
k
+
1
)
f
or
a
ll
1
≤
k
≤
2018.
Express your answer in the form
(
m
n
)
{m \choose n}
(
n
m
)
, where
m
m
m
and
n
n
n
are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.