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2014 MMATHS
2014 MMATHS
Part of
MMATHS problems
Subcontests
(6)
Mixer Round
1
Hide problems
2014 MMATHS Mixer Round - Math Majors of America Tournament for High Schools
p1. How many real roots does the equation
2
x
7
+
x
5
+
4
x
3
+
x
+
2
=
0
2x^7 + x^5 + 4x^3 + x + 2 = 0
2
x
7
+
x
5
+
4
x
3
+
x
+
2
=
0
have? p2. Given that
f
(
n
)
=
1
+
∑
j
=
1
n
(
1
+
∑
i
=
1
j
(
2
i
+
1
)
)
f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))
f
(
n
)
=
1
+
∑
j
=
1
n
(
1
+
∑
i
=
1
j
(
2
i
+
1
))
, find the value of
f
(
99
)
−
∑
i
=
1
99
i
2
f(99)-\sum^{99}_{i=1} i^2
f
(
99
)
−
∑
i
=
1
99
i
2
. p3. A rectangular prism with dimensions
1
×
a
×
b
1\times a \times b
1
×
a
×
b
, where
1
<
a
<
b
<
2
1 < a < b < 2
1
<
a
<
b
<
2
, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute
a
b
ab
ab
. Note: Two rectangular prisms of dimensions
p
×
q
×
r
p \times q\times r
p
×
q
×
r
and
x
×
y
×
z
x\times y\times z
x
×
y
×
z
are similar if
p
x
=
q
y
=
r
z
\frac{p}{x} = \frac{q}{y} = \frac{r}{z}
x
p
=
y
q
=
z
r
. p4. For fixed real values of
p
p
p
,
q
q
q
,
r
r
r
and
s
s
s
, the polynomial
x
4
+
p
x
3
+
q
x
2
+
r
x
+
s
x^4 + px^3 + qx^2 + rx + s
x
4
+
p
x
3
+
q
x
2
+
r
x
+
s
has four non real roots. The sum of two of these roots is
4
+
7
i
4 + 7i
4
+
7
i
, and the product of the other two roots is
3
−
4
i
3 - 4i
3
−
4
i
. Compute
q
q
q
. p5. There are
10
10
10
seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat
10
10
10
kids such that no two bad kids are allowed to sit next to each other? p6. There are an infinite number of people playing a game. They each pick a different positive integer
k
k
k
, and they each win the amount they chose with probability
1
k
3
\frac{1}{k^3}
k
3
1
. What is the expected amount that all of the people win in total? p7. There are
100
100
100
donuts to be split among
4
4
4
teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep? p8. Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear: Dominic: “Sitharthan is not credible.” Mitchell: “Dominic is not credible.” Sitharthan: “At least one of Dominic or Mitchell is credible.” Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!” They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible. p9. Pick an integer
n
n
n
between
1
1
1
and
10
10
10
. If no other team picks the same number, we’ll give you
n
10
\frac{n}{10}
10
n
points. p10. Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you
1
5
\frac15
5
1
points for each reasonable argument.
(
i
)
(
.
5
)
!
(
i
i
)
∞
⋅
0
(
i
i
i
)
0
0
(
i
v
)
∏
x
∈
∅
x
(
v
)
1
−
1
+
1
−
1
+
.
.
.
(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...
(
i
)
(
.5
)!
(
ii
)
∞
⋅
0
(
iii
)
0
0
(
i
v
)
x
∈
∅
∏
x
(
v
)
1
−
1
+
1
−
1
+
...
p11. On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
4
1
Hide problems
2014 MMATHS Tiebreaker p4 - min max n\sqrt5 - [ n\sqrt5]
Determine, with proof, the maximum and minimum among the numbers
5
−
⌊
5
⌋
,
2
5
−
⌊
2
5
⌋
,
3
5
−
⌊
3
5
⌋
,
.
.
.
,
2013
5
−
⌊
2013
5
⌋
,
2014
5
−
⌊
2014
5
⌋
\sqrt5 - \lfloor \sqrt5 \rfloor, 2\sqrt5 - \lfloor 2\sqrt5 \rfloor, 3\sqrt5 - \lfloor 3 \sqrt5\rfloor, ..., 2013\sqrt5 - \lfloor 2013\sqrt5\rfloor, 2014\sqrt5 - \lfloor 2014\sqrt5\rfloor
5
−
⌊
5
⌋
,
2
5
−
⌊
2
5
⌋
,
3
5
−
⌊
3
5
⌋
,
...
,
2013
5
−
⌊
2013
5
⌋
,
2014
5
−
⌊
2014
5
⌋
3
1
Hide problems
2014 MMATHS Tiebreaker p3 - f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}
Let
f
:
R
+
→
R
+
f : R^+ \to R^+
f
:
R
+
→
R
+
be a function satisfying
f
(
x
1
x
2
)
=
f
(
x
1
)
f
(
x
2
)
f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}
f
(
x
1
x
2
)
=
f
(
x
1
)
f
(
x
2
)
for all positive real numbers
x
1
,
x
2
x_1, x_2
x
1
,
x
2
. Show that
f
(
x
1
x
2
.
.
.
x
n
n
)
=
f
(
x
1
)
f
(
x
2
)
.
.
.
f
(
x
n
)
n
f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}
f
(
n
x
1
x
2
...
x
n
)
=
n
f
(
x
1
)
f
(
x
2
)
...
f
(
x
n
)
for all positive integers
n
n
n
and positive real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
.
2
1
Hide problems
2014 MMATHS Tiebreaker p2 - a_n = n if a_{2n} = a_n + n
Let
(
a
n
)
n
=
1
∞
(a_n)^{\infty}_{n =1}
(
a
n
)
n
=
1
∞
be a sequence of positive integers with
a
1
<
a
2
<
a
3
<
.
.
.
a_1 < a_2 < a_3 < ...
a
1
<
a
2
<
a
3
<
...
, and for n = 1, 2, 3,...,
a
2
n
=
a
n
+
n
.
a_{2n} = a_n + n.
a
2
n
=
a
n
+
n
.
Furthermore, whenever
n
n
n
is prime, so is
a
n
a_n
a
n
. Prove that
a
n
=
n
a_n = n
a
n
=
n
.
1
1
Hide problems
2014 MMATHS Tiebreaker p1 - integer right triangle with 1 odd side
Show that there does not exist a right triangle with all integer side lengths such that exactly one of the side lengths is odd.
1
Hide problems
2014 MMATHS Individual Round - Math Majors of America Tournament for High School
p1. For what value of
x
>
0
x > 0
x
>
0
does
f
(
x
)
=
(
x
−
3
)
2
(
x
+
4
)
f(x) = (x -3)^2(x + 4)
f
(
x
)
=
(
x
−
3
)
2
(
x
+
4
)
achieve the smallest value? p2. There are exactly
29
29
29
possible values that can be made using one or more of the
5
5
5
distinct coins with values
1
1
1
,
3
3
3
,
5
5
5
,
7
7
7
, and
X
X
X
. What is the smallest positive integral value for
X
X
X
? p3. Define
⋆
\star
⋆
as
x
⋆
y
=
x
−
1
x
y
x \star y = x -\frac{1}{xy}
x
⋆
y
=
x
−
x
y
1
. What is the sum of all complex
x
x
x
such that
x
⋆
(
x
⋆
2
x
)
=
2
x
x \star (x \star 2x) = 2x
x
⋆
(
x
⋆
2
x
)
=
2
x
? p4. Let
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
be the greatest integer less than or equal to
x
x
x
and let
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
be the least integer greater than or equal to
x
x
x
. Compute the smallest positive value of
a
a
a
for which
⌊
a
⌋
\lfloor a \rfloor
⌊
a
⌋
,
⌈
a
⌉
\lceil a \rceil
⌈
a
⌉
,
⌊
a
2
⌋
\lfloor a^2 \rfloor
⌊
a
2
⌋
is a nonconstant arithmetic sequence. p5. A right triangle is bounded in a coordinate plane by the lines
x
=
0
x = 0
x
=
0
,
y
=
0
y = 0
y
=
0
,
x
=
x
100
x = x_{100}
x
=
x
100
, and
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
, where
f
f
f
is a linear function with a negative slope and
f
(
x
100
)
=
0
f(x_{100}) = 0
f
(
x
100
)
=
0
. The lines
x
=
x
1
x = x_1
x
=
x
1
,
x
=
x
2
x = x_2
x
=
x
2
,
.
.
.
...
...
,
x
=
x
99
x = x_{99}
x
=
x
99
(
x
1
<
x
2
<
.
.
.
<
x
100
x_1 < x_2 <... < x_{100}
x
1
<
x
2
<
...
<
x
100
) subdivide the triangle into
100
100
100
regions of equal area. Compute
x
100
x
1
\frac{x_{100}}{x_1}
x
1
x
100
. p6. There are
10
10
10
children in a line to get candy. The pieces of candy are indistinguishable, while the children are not. If there are a total of
390
390
390
pieces of candy, how many ways are there to distribute the candy so that the
n
t
h
n^{th}
n
t
h
child in line receives at least
n
2
n^2
n
2
pieces of candy? p7. Compute
(
54
23
)
+
6
(
54
24
)
+
15
(
54
25
)
+
15
(
54
27
)
+
6
(
54
28
)
+
(
54
29
)
−
(
60
29
)
(
54
26
)
\frac{ {54 \choose 23}+ 6 {54 \choose 24}+ 15{54 \choose 25}+ 15{54 \choose 27}+ 6{54 \choose 28}+ {54 \choose 29} - {60 \choose 29}}{{54 \choose 26}}
(
26
54
)
(
23
54
)
+
6
(
24
54
)
+
15
(
25
54
)
+
15
(
27
54
)
+
6
(
28
54
)
+
(
29
54
)
−
(
29
60
)
p8. Point
A
A
A
lies on the circle centered at
O
O
O
.
A
B
‾
\overline{AB}
A
B
is tangent to
O
O
O
, and
C
C
C
is located on the circle so that
m
∠
A
O
C
=
12
0
o
m\angle AOC = 120^o
m
∠
A
OC
=
12
0
o
and oriented so that
∠
B
A
C
\angle BAC
∠
B
A
C
is obtuse.
B
C
‾
\overline{BC}
BC
intersects the circle at
D
D
D
. If
A
B
=
6
AB = 6
A
B
=
6
and
B
D
=
3
BD = 3
B
D
=
3
, then compute the radius of the circle. p9. The center of each face of a regular octahedron (a solid figure with
8
8
8
equilateral triangles as faces) with side length one unit is marked, and those points are the vertices of some cube. The center of each face of the cube is marked, and these points are the vertices of an even smaller regular octahedron. What is the volume of the smaller octahedron? p10. 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. p11. 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? p12. Define an operation
Φ
\Phi
Φ
whose input is a real-valued function and output is a real number so that it has the following properties:
∙
\bullet
∙
For any two real-valued functions
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
, and any real numbers
a
a
a
and
b
b
b
, then
Φ
(
a
f
(
x
)
+
b
g
(
x
)
)
=
a
Φ
(
f
(
x
)
)
+
b
Φ
(
g
(
x
)
)
\Phi (af(x) + bg(x)) = a \Phi (f(x)) + b\Phi (g(x))
Φ
(
a
f
(
x
)
+
b
g
(
x
))
=
a
Φ
(
f
(
x
))
+
b
Φ
(
g
(
x
))
∙
\bullet
∙
For any real-valued function
h
(
x
)
h(x)
h
(
x
)
, there is a polynomial function
p
(
x
)
p(x)
p
(
x
)
such that
Φ
(
p
(
x
)
⋅
h
(
x
)
)
=
Φ
(
(
h
(
x
)
)
2
)
\Phi (p(x) \cdot h(x)) = \Phi ((h(x))^2)
Φ
(
p
(
x
)
⋅
h
(
x
))
=
Φ
((
h
(
x
)
)
2
)
∙
\bullet
∙
If some function
m
(
x
)
m(x)
m
(
x
)
is always non-negative, and
Φ
(
m
(
x
)
)
=
0
\Phi (m(x)) = 0
Φ
(
m
(
x
))
=
0
, then
m
(
x
)
m(x)
m
(
x
)
is always
0
0
0
. Let
r
(
x
)
r(x)
r
(
x
)
be a real-valued function with
r
(
5
)
=
3
r(5) = 3
r
(
5
)
=
3
. Let
S
S
S
be the set of all real-valued functions
s
(
x
)
s(x)
s
(
x
)
that satisfy that
Φ
(
r
(
x
)
⋅
x
n
)
=
Φ
(
s
(
x
)
⋅
x
n
+
1
)
\Phi(r(x) \cdot x^n) = \Phi(s(x) \cdot x^{n+1})
Φ
(
r
(
x
)
⋅
x
n
)
=
Φ
(
s
(
x
)
⋅
x
n
+
1
)
. For each
s
s
s
in
S
S
S
, give the value of
s
(
5
)
s(5)
s
(
5
)
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.