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2015 MMATHS
2015 MMATHS
Part of
MMATHS problems
Subcontests
(6)
Mixer Round
1
Hide problems
2015 MMATHS Mixer Round - Math Majors of America Tournament for High Schools
p1. Let
a
0
,
a
1
,
.
.
.
,
a
n
a_0, a_1,...,a_n
a
0
,
a
1
,
...
,
a
n
be such that
a
n
≠
0
a_n \ne 0
a
n
=
0
and
(
1
+
x
+
x
3
)
341
(
1
+
2
x
+
x
2
+
2
x
3
+
2
x
4
+
x
6
)
342
=
∑
i
=
0
n
a
i
x
i
,
(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum^n_{i=0}a_ix^i,
(
1
+
x
+
x
3
)
341
(
1
+
2
x
+
x
2
+
2
x
3
+
2
x
4
+
x
6
)
342
=
i
=
0
∑
n
a
i
x
i
,
Find the number of odd numbers in the sequence a0; a1; : : : an. p2. Let
F
0
=
1
F_0 = 1
F
0
=
1
,
F
1
=
1
F_1 = 1
F
1
=
1
and F
k
=
F
k
−
1
+
F
k
−
2
_k = F_{k-1} + F_{k-2}
k
=
F
k
−
1
+
F
k
−
2
. Let
P
(
x
)
=
∑
k
=
0
99
x
F
k
P(x) =\sum^{99}_{k=0} x^{F_k}
P
(
x
)
=
∑
k
=
0
99
x
F
k
. The remainder when
P
(
x
)
P(x)
P
(
x
)
is divided by
x
3
−
1
x^3 - 1
x
3
−
1
can be expressed as
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
. Find
2
a
+
b
2a + b
2
a
+
b
. p3. Let
a
n
a_n
a
n
be the number of permutations of the numbers
S
=
{
1
,
2
,
.
.
.
,
n
}
S = \{1, 2,...,n\}
S
=
{
1
,
2
,
...
,
n
}
such that for all
k
k
k
with
1
≤
k
≤
n
1 \le k \le n
1
≤
k
≤
n
, the sum of
k
k
k
and the number in the
k
k
k
th position of the permutation is a power of
2
2
2
. Compute
a
2
0
+
a
2
1
+
.
.
.
+
a
2
20
a_{2^0} + a_{2^1} +... + a_{2^{20}}
a
2
0
+
a
2
1
+
...
+
a
2
20
. p4. Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color? p5. Compute the greatest positive integer
n
n
n
such that there exists an odd integer
a
a
a
, for which
a
2
n
−
1
4
4
4
\frac{a^{2^n}-1}{4^{4^4}}
4
4
4
a
2
n
−
1
is not an integer. p6. You are blind and cannot feel the difference between a coin that is heads up or tails up. There are
100
100
100
coins in front of you and are told that exactly
10
10
10
of them are heads up. On the back of this paper, explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads. p7. On the back of this page, write the best math pun you can think of. You’ll get a point if we chuckle. p8. Pick an integer between
1
1
1
and
10
10
10
. If you pick
k
k
k
, and
n
n
n
total teams pick
k
k
k
, then you’ll receive
k
10
n
\frac{k}{10n}
10
n
k
points. p9. There are four prisoners in a dungeon. Tomorrow, they will be separated into a group of three in one room, and the other in a room by himself. Each will be given a hat to wear that is either black or white – two will be given white and two black. None of them will be able to communicate with each other and none will see his or her own hat color. The group of three is lined up, so that the one in the back can see the other two, the second can see the first, but the first cannot see the others. If anyone is certain of their hat color, then they immediately shout that they know it to the rest of the group. If they can secretly prove it to the guard, they are saved. They only say something if they’re sure. Which person is sure to survive? p10. Down the road, there are
10
10
10
prisoners in a dungeon. Tomorrow they will be lined up in a single room and each given a black or white hat – this time they don’t know how many of each. The person in the back can see everyone’s hat besides his own, and similarly everyone else can only see the hats of the people in front of them. The person in the back will shout out a guess for his hat color and will be saved if and only if he is right. Then the person in front of him will have to guess, and this will continue until everyone has the opportunity to be saved. Each person can only say his or her guess of “white” or “black” when their turn comes, and no other signals may be made. If they have the night before receiving the hats to try to devise some sort of code, how many people at a minimum can be saved with the most optimal code? Describe the code on the back of this paper for full points. p11. A few of the problems on this mixer contest were taken from last year’s event. One of them had fewer than
5
5
5
correct answers, and most of the answers given were the same incorrect answer. Half a point will be given if you can guess the number of the problem on this test that corresponds to last year’s question, and another
.
5
.5
.5
points will be given if you can guess the very common incorrect answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
4
1
Hide problems
2015 MMATHS Tiebreaker p4 - n squirrels at a party, friendship among them
For any nonnegative integer
r
r
r
, let
S
r
S_r
S
r
be a function whose domain is the natural numbers that satisfies
S
r
(
p
α
)
=
{
0
i
f
i
f
p
≤
r
p
α
−
1
(
p
−
r
)
i
f
p
>
r
S_r(p^{\alpha}) = \begin{cases} 0\,\, if \,\, if \,\, p \le r \\ p^{{\alpha}-1}(p -r) \,\, if \,\,p > r \end{cases}
S
r
(
p
α
)
=
{
0
i
f
i
f
p
≤
r
p
α
−
1
(
p
−
r
)
i
f
p
>
r
for all primes
p
p
p
and positive integers
α
{\alpha}
α
, and that
S
r
(
a
b
)
=
S
r
(
a
)
S
r
(
b
)
S_r(ab) = S_r(a)Sr_(b)
S
r
(
ab
)
=
S
r
(
a
)
S
r
(
b
)
whenever
a
a
a
and
b
b
b
are relatively prime. Now, suppose there are
n
n
n
squirrels at a party. Each squirrel is labeled with a unique number from the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,..., n\}
{
1
,
2
,
...
,
n
}
. Two squirrels are friends with each other if and only if the difference between their labels is relatively prime to
n
n
n
. For example, if
n
=
10
n = 10
n
=
10
, then the squirrels with labels
3
3
3
and
10
10
10
are friends with each other because
10
−
3
=
7
10 - 3 = 7
10
−
3
=
7
, and
7
7
7
is relatively prime to
10
10
10
. Fix a positive integer
m
m
m
. Define a clique of size
m
m
m
to be any set of m squirrels at the party with the property that any two squirrels in the clique are friends with each other. Determine, with proof, a formula (using
S
r
S_r
S
r
) for the number of cliques of size
m
m
m
at the squirrel party.
3
1
Hide problems
2015 MMATHS Tiebreaker p3 - can first three digits of n\pi are .001 ?
Is there a number
s
s
s
in the set
{
π
,
2
π
,
3
π
,
.
.
.
,
}
\{\pi,2\pi,3\pi,...,\}
{
π
,
2
π
,
3
π
,
...
,
}
such that the first three digits after the decimal point of
s
s
s
are
.
001
.001
.001
? Fully justify your answer.
2
1
Hide problems
2015 MMATHS Tiebreaker p2 - is 22!6! + 1 prime?
Determine, with proof, whether
22
!
6
!
+
1
22!6! + 1
22
!
6
!
+
1
is prime.
1
1
Hide problems
2015 MMATHS Tiebreaker p1 - postitive integers label lattice points
Each lattice point of the plane is labeled by a positive integer. Each of these numbers is the arithmetic mean of its four neighbors (above, below, left, right). Show that all the numbers are equal.
3
Hide problems
2015 MMATHS Individual Round - Math Majors of America Tournament for High School
p1. The sum of two numbers,
a
+
b
a + b
a
+
b
, is
4
4
4
more than their difference,
a
−
b
a - b
a
−
b
. What is
b
b
b
? p2. Rectangle
A
B
C
D
ABCD
A
BC
D
has
A
B
=
C
D
=
3
AB = CD =\sqrt3
A
B
=
C
D
=
3
and
A
D
=
B
C
=
1
AD = BC = 1
A
D
=
BC
=
1
. Line
ℓ
\ell
ℓ
is the perpendicular bisector of
A
C
‾
\overline{AC}
A
C
and intersects
A
C
‾
\overline{AC}
A
C
and
A
B
‾
\overline{AB}
A
B
at
P
P
P
and
Q
Q
Q
, respectively. What is the area of quadrilateral
P
Q
B
C
PQBC
PQBC
? p3. The polynomial
p
(
x
)
=
x
3
+
c
x
−
2
p(x) = x^3 + cx - 2
p
(
x
)
=
x
3
+
c
x
−
2
has roots that are all integers. What is
c
c
c
? p4. There are
10
10
10
balls in a bucket, and there are
5
5
5
colors. Each color has exactly
2
2
2
balls of that color. Every time a ball is selected uniformly, randomly, and independently from the bucket, its color is noted and the ball is replaced. What is the expected number of selections from the bucket until one ball of every color has been seen? p5. Consider a solid rectangular prism with length
10
10
10
, width
8
8
8
, and height
6
6
6
. Find the volume of the set of points that are both a distance of at most
3
3
3
from the prism and a distance of at least
1
1
1
from the prism. p6. Two positive integers,
a
a
a
and
b
b
b
are chosen randomly, uniformly, and independently from the set of positive integers less than
1000
1000
1000
,
{
1
,
2
,
.
.
.
,
1000
}
\{1, 2,...,1000\}
{
1
,
2
,
...
,
1000
}
. What is the expected value of the number of quadrants through which the graph
x
a
+
y
b
=
1
x^a + y^b = 1
x
a
+
y
b
=
1
will pass? p7. John and Mitchell are playing a game to see who gets the last of their candy. John rolls three unbiased
7
7
7
-sided dice with sides labeled
1
1
1
through
7
7
7
and records the maximum roll. Mitchell flips a biased coin (with probability of heads
p
p
p
)
10
10
10
times and records the number of heads. What is the smallest
p
p
p
such that Mitchell has
50
%
50\%
50%
or greater chance of winning the candy by recording a larger number? p8. Define the sum of a finite set of integers to be the sum of the elements of the set. Let
D
D
D
be the set of positive divisors of
700
700
700
. How many nonempty subsets of
D
D
D
have an even sum? (Simplify as reasonably as possible) p9. Compute the absolute minimum of the function
f
(
x
)
=
cos
(
2
x
)
+
3
cos
(
x
)
f(x) = \cos (2x) + 3 \cos(x)
f
(
x
)
=
cos
(
2
x
)
+
3
cos
(
x
)
p10. The largest prime factor of the number
520
520
520
,
302
302
302
,
325
325
325
has
5
5
5
digits. What is this prime factor? p11. We play the following game with an equilateral triangle of
n
(
n
+
1
)
2
\frac{n(n+1)}{2}
2
n
(
n
+
1
)
coins (with
n
>
1
n > 1
n
>
1
coins on each side). Initially, all of the coins are turned heads up. On each turn, we may turn over three coins that are mutually adjacent; the goal is to make all of the coins eventually turned tails up. What is the
7
t
h
7^{th}
7
t
h
smallest positive
n
n
n
for which this can be achieved? p12. Let
S
=
{
a
1
,
a
2
,
a
3
,
.
.
.
,
a
2015
}
S = \{a_1, a_2, a_3,..., a_{2015}\}
S
=
{
a
1
,
a
2
,
a
3
,
...
,
a
2015
}
be the first
2015
2015
2015
positive integers that
a
a
a
can be so that
2
+
2
28
a
2
+
1
2 + 2\sqrt{28a^2 + 1}
2
+
2
28
a
2
+
1
is an integer. Compute the number of elements in
S
S
S
that a can be so that
2
+
2
28
a
2
+
1
2 + 2\sqrt{28a^2 + 1}
2
+
2
28
a
2
+
1
is not a perfect square. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2015 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS
Round 1 p1. If this mathathon has
7
7
7
rounds of
3
3
3
problems each, how many problems does it have in total? (Not a trick!) p2. Five people, named
A
,
B
,
C
,
D
,
A, B, C, D,
A
,
B
,
C
,
D
,
and
E
E
E
, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? p3. At Barrios’s absurdly priced fish and chip shop, one fish is worth
$
13
\$13
$13
, one chip is worth
$
5
\$5
$5
. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? Round 2 p4. If there are
15
15
15
points in
4
4
4
-dimensional space, what is the maximum number of hyperplanes that these points determine? p5. Consider all possible values of
z
1
−
z
2
z
2
−
z
3
⋅
z
1
−
z
4
z
2
−
z
4
\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}
z
2
−
z
3
z
1
−
z
2
⋅
z
2
−
z
4
z
1
−
z
4
for any distinct complex numbers
z
1
z_1
z
1
,
z
2
z_2
z
2
,
z
3
z_3
z
3
, and
z
4
z_4
z
4
. How many complex numbers cannot be achieved? p6. For each positive integer
n
n
n
, let
S
(
n
)
S(n)
S
(
n
)
denote the number of positive integers
k
≤
n
k \le n
k
≤
n
such that
g
c
d
(
k
,
n
)
=
g
c
d
(
k
+
1
,
n
)
=
1
gcd(k, n) = gcd(k + 1, n) = 1
g
c
d
(
k
,
n
)
=
g
c
d
(
k
+
1
,
n
)
=
1
. Find
S
(
2015
)
S(2015)
S
(
2015
)
. Round 3 p7. Let
P
1
P_1
P
1
,
P
2
P_2
P
2
,
.
.
.
...
...
,
P
2015
P_{2015}
P
2015
be
2015
2015
2015
distinct points in the plane. For any
i
,
j
∈
{
1
,
2
,
.
.
.
.
,
2015
}
i, j \in \{1, 2, ...., 2015\}
i
,
j
∈
{
1
,
2
,
....
,
2015
}
, connect
P
i
P_i
P
i
and
P
j
P_j
P
j
with a line segment if and only if
g
c
d
(
i
−
j
,
2015
)
=
1
gcd(i - j, 2015) = 1
g
c
d
(
i
−
j
,
2015
)
=
1
. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let
ω
\omega
ω
be the unique positive integer such that there exists a clique with
ω
\omega
ω
elements and such that there does not exist a clique with
ω
+
1
\omega + 1
ω
+
1
elements. Find
ω
\omega
ω
. p8. A Chinese restaurant has many boxes of food. The manager notices that
∙
\bullet
∙
He can divide the boxes into groups of
M
M
M
where
M
M
M
is
19
19
19
,
20
20
20
, or
21
21
21
.
∙
\bullet
∙
There are exactly
3
3
3
integers
x
x
x
less than
16
16
16
such that grouping the boxes into groups of
x
x
x
leaves
3
3
3
boxes left over. Find the smallest possible number of boxes of food. p9. If
f
(
x
)
=
x
∣
x
∣
+
2
f(x) = x|x| + 2
f
(
x
)
=
x
∣
x
∣
+
2
, then compute
∑
k
=
−
1000
1000
f
−
1
(
f
(
k
)
+
f
(
−
k
)
+
f
−
1
(
k
)
)
\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))
∑
k
=
−
1000
1000
f
−
1
(
f
(
k
)
+
f
(
−
k
)
+
f
−
1
(
k
))
. Round 4 p10. Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
20
BC = 20
BC
=
20
,
C
A
=
21
CA = 21
C
A
=
21
. Let
A
B
D
E
ABDE
A
B
D
E
,
B
C
F
G
BCFG
BCFG
, and
C
A
H
I
CAHI
C
A
H
I
be squares built on sides
A
B
AB
A
B
,
B
C
BC
BC
, and
C
A
CA
C
A
, respectively such that these squares are outside of
A
B
C
ABC
A
BC
. Find the area of
D
E
H
I
F
G
DEHIFG
D
E
H
I
FG
. p11. What is the sum of all of the distinct prime factors of
7783
=
6
5
+
6
+
1
7783 = 6^5 + 6 + 1
7783
=
6
5
+
6
+
1
? p12. Consider polyhedron
A
B
C
D
E
ABCDE
A
BC
D
E
, where
A
B
C
D
ABCD
A
BC
D
is a regular tetrahedron and
B
C
D
E
BCDE
BC
D
E
is a regular tetrahedron. An ant starts at point
A
A
A
. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After
6
6
6
moves, what is the probability the ant is back at point
A
A
A
? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2015 MMATHS Mathathon Rounds 5-7 Math Majors of America Tournament for HS
Round 5 p13. You have a
26
×
26
26 \times 26
26
×
26
grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of
2
×
2
2 \times 2
2
×
2
squares that are entirely red? p14. Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? p15. Let
n
≥
1
n \ge 1
n
≥
1
be an integer and
a
>
0
a > 0
a
>
0
a real number. In terms of n, find the number of solutions
(
x
1
,
.
.
.
,
x
n
)
(x_1, ..., x_n)
(
x
1
,
...
,
x
n
)
of the equation
∑
i
=
1
n
(
x
i
2
+
(
a
−
x
i
)
2
)
=
n
a
2
\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2
∑
i
=
1
n
(
x
i
2
+
(
a
−
x
i
)
2
)
=
n
a
2
such that
x
i
x_i
x
i
belongs to the interval
[
0
,
a
]
[0, a]
[
0
,
a
]
, for
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2, . . . , n
i
=
1
,
2
,
...
,
n
. Round 6 p16. All roots of
∏
n
=
1
25
∏
k
=
0
2
n
(
−
1
)
k
⋅
x
k
=
0
\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0
n
=
1
∏
25
k
=
0
∏
2
n
(
−
1
)
k
⋅
x
k
=
0
are written in the form
r
(
cos
ϕ
+
i
sin
ϕ
)
r(\cos \phi + i\sin \phi)
r
(
cos
ϕ
+
i
sin
ϕ
)
for
i
2
=
−
1
i^2 = -1
i
2
=
−
1
,
r
>
0
r > 0
r
>
0
, and
0
≤
ϕ
<
2
π
0 \le \phi < 2\pi
0
≤
ϕ
<
2
π
. What is the smallest positive value of
ϕ
\phi
ϕ
in radians? p17. Find the sum of the distinct real roots of the equation
x
2
−
2
x
+
1
3
+
x
2
−
x
−
6
3
=
2
x
2
−
3
x
−
5
3
.
\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.
3
x
2
−
2
x
+
1
+
3
x
2
−
x
−
6
=
3
2
x
2
−
3
x
−
5
.
p18. If
a
a
a
and
b
b
b
satisfy the property that
a
2
n
+
b
a2^n + b
a
2
n
+
b
is a square for all positive integers
n
n
n
, find all possible value(s) of
a
a
a
. Round 7 p19. Compute
(
1
−
cot
1
9
o
)
(
1
−
cot
2
6
o
)
(1 - \cot 19^o)(1 - \cot 26^o)
(
1
−
cot
1
9
o
)
(
1
−
cot
2
6
o
)
. p20. Consider triangle
A
B
C
ABC
A
BC
with
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
5
BC = 5
BC
=
5
, and
∠
A
B
C
=
12
0
o
\angle ABC = 120^o
∠
A
BC
=
12
0
o
. Let point
E
E
E
be any point inside
A
B
C
ABC
A
BC
. The minimum of the sum of the squares of the distances from
E
E
E
to the three sides of
A
B
C
ABC
A
BC
can be written in the form
a
/
b
a/b
a
/
b
, where a and b are natural numbers such that the greatest common divisor of
a
a
a
and
b
b
b
is
1
1
1
. Find
a
+
b
a + b
a
+
b
. p21. Let
m
≠
1
m \ne 1
m
=
1
be a square-free number (an integer – possibly negative – such that no square divides
m
m
m
). We denote
Q
(
m
)
Q(\sqrt{m})
Q
(
m
)
to be the set of all
a
+
b
m
a + b\sqrt{m}
a
+
b
m
where
a
a
a
and
b
b
b
are rational numbers. Now for a fixed
m
m
m
, let
S
S
S
be the set of all numbers
x
x
x
in
Q
(
m
)
Q(\sqrt{m})
Q
(
m
)
such that x is a solution to a polynomial of the form:
x
n
+
a
1
x
n
−
1
+
.
.
.
.
+
a
n
=
0
x^n + a_1x^{n-1} + .... + a_n = 0
x
n
+
a
1
x
n
−
1
+
....
+
a
n
=
0
, where
a
0
a_0
a
0
,
.
.
.
...
...
,
a
n
a_n
a
n
are integers. For many integers m,
S
=
Z
[
m
]
=
{
a
+
b
m
}
S = Z[\frac{m}] = \{a + b\sqrt{m}\}
S
=
Z
[
]
m
=
{
a
+
b
m
}
where
a
a
a
and
b
b
b
are integers. Give a classification of the integers for which this is not true. (Hint: It is true for
m
=
−
1
m = -1
m
=
−
1
and
2
2
2
.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.