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Duke Math Meet (DMM)
2016 Duke Math Meet
2016 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
(1)
3
Hide problems
2016 DMM Tiebreaker Round - Duke Math Meet
p1. How many ordered triples of integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
where
1
≤
a
,
b
,
c
≤
10
1 \le a, b, c \le 10
1
≤
a
,
b
,
c
≤
10
are such that for every natural number, the equation
(
a
+
n
)
x
2
+
(
b
+
2
n
)
x
+
c
+
n
=
0
(a + n)x^2 + (b + 2n)x + c + n = 0
(
a
+
n
)
x
2
+
(
b
+
2
n
)
x
+
c
+
n
=
0
has at least one real root? p2. Find the smallest integer
n
n
n
such that we can cut a
n
×
n
n \times n
n
×
n
grid into
5
5
5
rectangles with distinct side lengths in
{
1
,
2
,
3...
,
10
}
\{1, 2, 3..., 10\}
{
1
,
2
,
3...
,
10
}
. Every value is used exactly once. p3. A plane is flying at constant altitude along a circle of radius
12
12
12
miles with center at a point
A
A
A
.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point
A
A
A
, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2016 DMM Individual Round - Duke Math Meet
p1. Trung took five tests this semester. For his first three tests, his average was
60
60
60
, and for the fourth test he earned a
50
50
50
. What must he have earned on his fifth test if his final average for all five tests was exactly
60
60
60
? p2. Find the number of pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
20
a
+
16
b
=
2016
−
a
b
20a + 16b = 2016 - ab
20
a
+
16
b
=
2016
−
ab
. p3. Let
f
:
N
→
N
f : N \to N
f
:
N
→
N
be a strictly increasing function with
f
(
1
)
=
2016
f(1) = 2016
f
(
1
)
=
2016
and
f
(
2
t
)
=
f
(
t
)
+
t
f(2t) = f(t) + t
f
(
2
t
)
=
f
(
t
)
+
t
for all
t
∈
N
t \in N
t
∈
N
. Find
f
(
2016
)
f(2016)
f
(
2016
)
. p4. Circles of radius
7
7
7
,
7
7
7
,
18
18
18
, and
r
r
r
are mutually externally tangent, where
r
=
m
n
r = \frac{m}{n}
r
=
n
m
for relatively prime positive integers
m
m
m
and
n
n
n
. Find
m
+
n
m + n
m
+
n
. p5. A point is chosen at random from within the circumcircle of a triangle with angles
4
5
o
45^o
4
5
o
,
7
5
o
75^o
7
5
o
,
6
0
o
60^o
6
0
o
. What is the probability that the point is closer to the vertex with an angle of
4
5
o
45^o
4
5
o
than either of the two other vertices? p6. Find the largest positive integer
a
a
a
less than
100
100
100
such that for some positive integer
b
b
b
,
a
−
b
a - b
a
−
b
is a prime number and
a
b
ab
ab
is a perfect square. p7. There is a set of
6
6
6
parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds
6
6
6
more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. p8. Triangle
A
B
C
ABC
A
BC
has sides
A
B
=
5
AB = 5
A
B
=
5
,
A
C
=
4
AC = 4
A
C
=
4
, and
B
C
=
3
BC = 3
BC
=
3
. Let
O
O
O
be any arbitrary point inside
A
B
C
ABC
A
BC
, and
D
∈
B
C
D \in BC
D
∈
BC
,
E
∈
A
C
E \in AC
E
∈
A
C
,
F
∈
A
B
F \in AB
F
∈
A
B
, such that
O
D
⊥
B
C
OD \perp BC
O
D
⊥
BC
,
O
E
⊥
A
C
OE \perp AC
OE
⊥
A
C
,
O
F
⊥
A
B
OF \perp AB
OF
⊥
A
B
. Find the minimum value of
O
D
2
+
O
E
2
+
O
F
2
OD^2 + OE^2 + OF^2
O
D
2
+
O
E
2
+
O
F
2
. p9. Find the root with the largest real part to
x
4
−
3
x
3
+
3
x
+
1
=
0
x^4-3x^3+3x+1 = 0
x
4
−
3
x
3
+
3
x
+
1
=
0
over the complex numbers. p10. Tony has a board with
2
2
2
rows and
4
4
4
columns. Tony will use
8
8
8
numbers from
1
1
1
to
8
8
8
to fill in this board, each number in exactly one entry. Let array
(
a
1
,
.
.
.
,
a
4
)
(a_1,..., a_4)
(
a
1
,
...
,
a
4
)
be the first row of the board and array
(
b
1
,
.
.
.
,
b
4
)
(b_1,..., b_4)
(
b
1
,
...
,
b
4
)
be the second row of the board. Let
F
=
∑
i
=
1
4
∣
a
i
−
b
i
∣
F =\sum^{4}_{i=1}|a_i - b_i|
F
=
∑
i
=
1
4
∣
a
i
−
b
i
∣
, calculate the average value of
F
F
F
across all possible ways to fill in. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2016 DMM Team Round - Duke Math Meet
p1. What is the maximum number of
T
T
T
-shaped polyominos (shown below) that we can put into a
6
×
6
6 \times 6
6
×
6
grid without any overlaps. The blocks can be rotated. https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png p2. In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,
∠
A
=
3
0
o
\angle A = 30^o
∠
A
=
3
0
o
.
D
D
D
is a point on
A
B
AB
A
B
such that
C
D
⊥
A
B
CD \perp AB
C
D
⊥
A
B
.
E
E
E
is a point on
A
C
AC
A
C
such that
B
E
⊥
A
C
BE \perp AC
BE
⊥
A
C
. What is the value of
D
E
B
C
\frac{DE}{BC}
BC
D
E
? p3. Given that f(x) is a polynomial such that
2
f
(
x
)
+
f
(
1
−
x
)
=
x
2
2f(x) + f(1 - x) = x^2
2
f
(
x
)
+
f
(
1
−
x
)
=
x
2
. Find the sum of squares of the coefficients of
f
(
x
)
f(x)
f
(
x
)
. p4. For each positive integer
n
n
n
, there exists a unique positive integer an such that
a
n
2
≤
n
<
(
a
n
+
1
)
2
a^2_n \le n < (a_n + 1)^2
a
n
2
≤
n
<
(
a
n
+
1
)
2
. Given that
n
=
15
m
2
n = 15m^2
n
=
15
m
2
, where
m
m
m
is a positive integer greater than
1
1
1
. Find the minimum possible value of
n
−
a
n
2
n - a^2_n
n
−
a
n
2
. p5. What are the last two digits of
⌊
(
5
+
2
)
2016
⌋
\lfloor (\sqrt5 + 2)^{2016}\rfloor
⌊(
5
+
2
)
2016
⌋
? Note
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
is the largest integer less or equal to x. p6. Let
f
f
f
be a function that satisfies
f
(
2
a
3
b
)
)
=
3
a
+
5
b
f(2^a3^b)) = 3a+ 5b
f
(
2
a
3
b
))
=
3
a
+
5
b
. What is the largest value of f over all numbers of the form
n
=
2
a
3
b
n = 2^a3^b
n
=
2
a
3
b
where
n
≤
10000
n \le 10000
n
≤
10000
and
a
,
b
a, b
a
,
b
are nonnegative integers. p7. Find a multiple of
21
21
21
such that it has six more divisors of the form
4
m
+
1
4m + 1
4
m
+
1
than divisors of the form
4
n
+
3
4n + 3
4
n
+
3
where m, n are integers. You can keep the number in its prime factorization form. p8. Find
∑
i
=
0
100
⌊
i
3
/
2
⌋
+
∑
j
=
0
1000
⌊
j
2
/
3
⌋
\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor
i
=
0
∑
100
⌊
i
3/2
⌋
+
j
=
0
∑
1000
⌊
j
2/3
⌋
where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
is the largest integer less or equal to x. p9. Let
A
,
B
A, B
A
,
B
be two randomly chosen subsets of
{
1
,
2
,
.
.
.
10
}
\{1, 2, . . . 10\}
{
1
,
2
,
...10
}
. What is the probability that one of the two subsets contains the other? p10. We want to pick
5
5
5
-person teams from a total of
m
m
m
people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint:
m
m
m
is determined by these conditions). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.