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Duke Math Meet (DMM)
2015 Duke Math Meet
2015 Duke Math Meet
Part of
Duke Math Meet (DMM)
Subcontests
(1)
2
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2015 DMM Individual Round - Duke Math Meet
p1. Find the minimum value of
x
4
+
2
x
3
+
3
x
2
+
2
x
+
2
x^4 +2x^3 +3x^2 +2x+2
x
4
+
2
x
3
+
3
x
2
+
2
x
+
2
, where x can be any real number. p2. A type of digit-lock has
5
5
5
digits, each digit chosen from
{
1
,
2
,
3
,
4
,
5
}
\{1,2, 3, 4, 5\}
{
1
,
2
,
3
,
4
,
5
}
. How many different passwords are there that have an odd number of
1
1
1
's? p3. Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have
80
%
80\%
80%
chance to be good balls and MIO balls have
50
%
50\%
50%
chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball? p4. What is the smallest positive integer that is one-ninth of its reverse? p5. When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many di erent orders can Michael get dressed? p6. The numbers
1
,
2
,
.
.
.
,
2015
1, 2,..., 2015
1
,
2
,
...
,
2015
are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number? p7. Let
A
=
(
a
1
b
1
a
2
b
2
.
.
.
a
n
b
n
)
34
A = (a_1b_1a_2b_2... a_nb_n)_{34}
A
=
(
a
1
b
1
a
2
b
2
...
a
n
b
n
)
34
and
B
=
(
b
1
b
2
.
.
.
b
n
)
34
B = (b_1b_2... b_n)_{34}
B
=
(
b
1
b
2
...
b
n
)
34
be two numbers written in base
34
34
34
. If the sum of the base-
34
34
34
digits of
A
A
A
is congruent to
15
15
15
(mod
77
77
77
) and the sum of the base
34
34
34
digits of
B
B
B
is congruent to
23
23
23
(mod
77
77
77
). Then if
(
a
1
b
1
a
2
b
2
.
.
.
a
n
b
n
)
34
≡
x
(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x
(
a
1
b
1
a
2
b
2
...
a
n
b
n
)
34
≡
x
(mod
77
77
77
) and
0
≤
x
≤
76
0 \le x \le 76
0
≤
x
≤
76
, what is
x
x
x
? (you can write
x
x
x
in base
10
10
10
) p8. What is the sum of the medians of all nonempty subsets of
{
1
,
2
,
.
.
.
,
9
}
\{1, 2,..., 9\}
{
1
,
2
,
...
,
9
}
? p9. Tony is moving on a straight line for
6
6
6
minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes? p10. Find the number of nonnegative integer triplets
a
,
b
,
c
a, b, c
a
,
b
,
c
that satisfy
2
a
3
b
+
9
=
c
2
.
2^a3^b + 9 = c^2.
2
a
3
b
+
9
=
c
2
.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2015 DMM Team Round - Duke Math Meet
p1. Let
U
=
{
−
2
,
0
,
1
}
U = \{-2, 0, 1\}
U
=
{
−
2
,
0
,
1
}
and
N
=
{
1
,
2
,
3
,
4
,
5
}
N = \{1, 2, 3, 4, 5\}
N
=
{
1
,
2
,
3
,
4
,
5
}
. Let
f
f
f
be a function that maps
U
U
U
to
N
N
N
. For any
x
∈
U
x \in U
x
∈
U
,
x
+
f
(
x
)
+
x
f
(
x
)
x + f(x) + xf(x)
x
+
f
(
x
)
+
x
f
(
x
)
is an odd number. How many
f
f
f
satisfy the above statement? p2. Around a circle are written all of the positive integers from
1
1
1
to
n
n
n
,
n
≥
2
n \ge 2
n
≥
2
in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest
n
n
n
for which this is possible. p3. Michael loses things, especially his room key. If in a day of the week he has
n
n
n
classes he loses his key with probability
n
/
5
n/5
n
/5
. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has
2
2
2
classes and that on Wednesday and Friday he has
3
3
3
classes, what is the probability that loses his key at least once during a week? p4. Given two concentric circles one with radius
8
8
8
and the other
5
5
5
. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than
7
7
7
? p5. We say that a positive integer
n
n
n
is lucky if
n
2
n^2
n
2
can be written as the sum of
n
n
n
consecutive positive integers. Find the number of lucky numbers strictly less than
2015
2015
2015
. p6. Let
A
=
{
3
x
+
3
y
+
3
z
∣
x
,
y
,
z
≥
0
,
x
,
y
,
z
∈
Z
,
x
<
y
<
z
}
A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}
A
=
{
3
x
+
3
y
+
3
z
∣
x
,
y
,
z
≥
0
,
x
,
y
,
z
∈
Z
,
x
<
y
<
z
}
. Arrange the set
A
A
A
in increasing order. Then what is the
50
50
50
th number? (Express the answer in the form
3
x
+
3
y
+
3
z
3^x + 3^y + 3^z
3
x
+
3
y
+
3
z
). p7. Justin and Oscar found
2015
2015
2015
sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win? p8. Let
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
with
x
≥
y
≥
z
≥
0
x \ge y \ge z \ge 0
x
≥
y
≥
z
≥
0
be integers such that
x
3
+
y
3
+
z
3
3
=
x
y
z
+
21
\frac{x^3+y^3+z^3}{3} = xyz + 21
3
x
3
+
y
3
+
z
3
=
x
yz
+
21
. Find
x
x
x
. p9. Let
p
<
q
<
r
<
s
p < q < r < s
p
<
q
<
r
<
s
be prime numbers such that
1
−
1
p
−
1
q
−
1
r
−
1
s
=
1
p
q
r
s
.
1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.
1
−
p
1
−
q
1
−
r
1
−
s
1
=
pq
rs
1
.
Find
p
+
q
+
r
+
s
p + q + r + s
p
+
q
+
r
+
s
. p10. In ”island-land”, there are
10
10
10
islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.