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Stanford Mathematics Tournament
2008 Stanford Mathematics Tournament
2008 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(16)
1
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2008 RMT + SMT Team Round - Rice + Stanford Math Tournament
p1. Find the maximum value of
e
sin
x
cos
x
tan
x
e^{\sin x \cos x \tan x}
e
s
i
n
x
c
o
s
x
t
a
n
x
. p2. A fighter pilot finds that the average number of enemy ZIG planes she shoots down is
56
z
−
4
z
2
56z -4z^2
56
z
−
4
z
2
, where
z
z
z
is the number of missiles she fires. Intending to maximize the number of planes she shoots down, she orders her gunner to “Have a nap ... then fire
z
z
z
missiles!” where
z
z
z
is an integer. What should
z
z
z
be? p3. A sequence is generated as follows: if the
n
t
h
n^{th}
n
t
h
term is even, then the
(
n
+
1
)
t
h
(n+ 1)^{th}
(
n
+
1
)
t
h
term is half the
n
t
h
n^{th}
n
t
h
term; otherwise it is two more than twice the
n
t
h
n^{th}
n
t
h
term. If the first term is
10
10
10
, what is the
200
8
t
h
2008^{th}
200
8
t
h
term? p4. Find the volume of the solid formed by rotating the area under the graph of
y
=
x
y =\sqrt{x}
y
=
x
around the
x
x
x
-axis, with
0
≤
x
≤
2
0 \le x \le 2
0
≤
x
≤
2
. p5. Find the volume of a regular octahedron whose vertices are at the centers of the faces of a unit cube. p6. What is the area of the triangle with vertices
(
x
,
0
,
0
)
(x, 0, 0)
(
x
,
0
,
0
)
,
(
0
,
y
,
0
)
(0, y, 0)
(
0
,
y
,
0
)
, and
(
0
,
0
,
z
)
(0, 0, z)
(
0
,
0
,
z
)
? p7. Daphne is in a maze of tunnels shown below. She enters at
A
A
A
, and at each intersection, chooses a direction randomly (including possibly turning around). Once Daphne reaches an exit, she will not return into the tunnels. What is the probability that she will exit at
A
A
A
? https://cdn.artofproblemsolving.com/attachments/c/0/0f8777e9ac9cbe302f042d040e8864d68cadb6.png p8. In triangle
A
X
E
AXE
A
XE
,
T
T
T
is the midpoint of
E
X
‾
\overline{EX}
EX
, and
P
P
P
is the midpoint of
E
T
‾
\overline{ET}
ET
. If triangle
A
P
E
APE
A
PE
is equilateral, find
cos
(
m
∠
X
A
E
)
\cos(m \angle XAE)
cos
(
m
∠
X
A
E
)
. p9. In rectangle
X
K
C
D
XKCD
X
K
C
D
,
J
J
J
lies on
K
C
‾
\overline{KC}
K
C
and
Z
Z
Z
lies on
X
K
‾
\overline{XK}
X
K
. If
X
J
‾
\overline{XJ}
X
J
and
K
D
‾
\overline{KD}
KD
intersect at
Q
Q
Q
,
Q
Z
‾
⊥
X
K
‾
\overline{QZ} \perp \overline{XK}
QZ
⊥
X
K
, and
K
C
K
J
=
n
\frac{KC}{KJ} = n
K
J
K
C
=
n
, find
X
D
Q
Z
\frac{XD}{QZ}
QZ
X
D
. p10. Bill the magician has cards
A
A
A
,
B
B
B
, and
C
C
C
as shown. For his act, he asks a volunteer to pick any number from
1
1
1
through
8
8
8
and tell him which cards among
A
A
A
,
B
B
B
, and
C
C
C
contain it. He then uses this information to guess the volunteer’s number (for example, if the volunteer told Bill “
A
A
A
and
C
C
C
”, he would guess “3”). One day, Bill loses card
C
C
C
and cannot remember which numbers were on it. He is in a hurry and randomly chooses four different numbers from
1
1
1
to
8
8
8
to write on a card. What is the probability Bill will still be able to do his trick?
A
A
A
: 2 3 5 7
B
B
B
: 2 4 6 7
C
C
C
: 2 3 6 1 p11. Given that
f
(
x
,
y
)
=
x
7
y
8
+
x
4
y
14
+
A
f(x, y) = x^7y^8 + x^4y^{14} + A
f
(
x
,
y
)
=
x
7
y
8
+
x
4
y
14
+
A
has root
(
16
,
7
)
(16, 7)
(
16
,
7
)
, find another root. p12. How many nonrectangular trapezoids can be formed from the vertices of a regular octagon? p13. If
r
e
i
θ
re^{i\theta}
r
e
i
θ
is a root of
x
8
−
x
7
+
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
1
=
0
x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = 0
x
8
−
x
7
+
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
1
=
0
,
r
>
0
r > 0
r
>
0
, and
0
≤
θ
<
360
0 \le \theta < 360
0
≤
θ
<
360
with
θ
\theta
θ
in degrees, find all possible values of
θ
\theta
θ
. p14. For what real values of
n
n
n
is
∫
−
π
2
π
2
(
tan
(
x
)
)
n
d
x
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\tan(x))^n dx
∫
−
2
π
2
π
(
tan
(
x
)
)
n
d
x
defined? p15. A parametric graph is given by
{
y
=
sin
t
x
=
cos
t
+
1
2
t
\begin{cases} y = \sin t \\ x = \cos t +\frac12 t \end{cases}
{
y
=
sin
t
x
=
cos
t
+
2
1
t
How many times does the graph intersect itself between
x
=
1
x = 1
x
=
1
and
x
=
40
x = 40
x
=
40
? PS. You had better use hide for answers.
16
1
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Area of a Triangle in a Hexagon
Suppose convex hexagon
HEXAGN
\text{HEXAGN}
HEXAGN
has
12
0
∘
120^\circ
12
0
∘
-rotational symmetry about a point
P
P
P
—that is, if you rotate it
12
0
∘
120^\circ
12
0
∘
about
P
P
P
, it doesn't change. If PX\equal{}1, find the area of triangle
△
G
H
X
\triangle{GHX}
△
G
H
X
.
15
1
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Death Run
While out for a stroll, you encounter a vicious velociraptor. You start running away to the northeast at
10
m/s
10 \text{m/s}
10
m/s
, and you manage a three-second head start over the raptor. If the raptor runs at
15
2
m/s
15\sqrt{2} \text{m/s}
15
2
m/s
, but only runs either north or east at any given time, how many seconds do you have until it devours you?
14
1
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A World of Natural Birth Control
Suppose families always have one, two, or three children, with probability ¼, ½, ¼ respectively. Assuming everyone eventually gets married and has children, what is the probability of a couple having exactly four grandchildren?
13
1
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Supercalifragilisticexpialidocious
Let N be the number of distinct rearrangements of the 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS. How many positive factors does N have?
11
1
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Messy Radicals
Simplify: \sqrt [3]{\frac {17\sqrt7 \plus{} 45}{4}}
10
1
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Very Ingenious Game
Six people play the following game: They have a cube, initially white. One by one, the players mark an
X
X
X
on a white face of the cube, and roll it like a die. The winner is the first person to roll an
X
X
X
(for example, player 1 wins with probability
1
6
\frac {1}{6}
6
1
, while if none of players 1-5 win, player 6 will place an
X
X
X
on the last square and win for sure). What is the probability that the sixth player wins?
9
1
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Factorial and Prime Factors
What is the sum of the prime factors of 20!?
8
1
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Distracted by Thoughts of Greater Magnitude
Terence Tao is playing rock-paper-scissors. Because his mental energy is focused on solving the twin primes conjecture, he uses the following very simple strategy: ·He plays rock first. ·On each subsequent turn, he plays a different move than the previous one, each with probability ½. What is the probability that his 5th move will be rock?
7
1
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Conditional Probability
At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?
6
1
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Sharpening a Pencil
A round pencil has length
8
8
8
when unsharpened, and diameter
1
4
\frac {1}{4}
4
1
. It is sharpened perfectly so that it remains
8
8
8
inches long, with a
7
7
7
inch section still cylindrical and the remaining
1
1
1
inch giving a conical tip. What is its volume?
5
1
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Fahrenheit and Celsius
One day, the temperature increases steadily from a low of
4
5
∘
F
45^\circ \text{F}
4
5
∘
F
in the early morning to a high of
7
0
∘
F
70^\circ \text{F}
7
0
∘
F
in the late afternoon. At how many times from early morning to late afternoon was the temperature an integer in both Fahrenheit and Celsius? Recall that C \equal{} \frac {5}{9}(F \minus{} 32).
4
1
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Right Triangle
A right triangle has sides of integer length. One side has length 11. What is the area of the triangle?
3
1
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Roots
Give the positive root(s) of x^3 \plus{} 2x^2 \minus{} 2x \minus{} 4.
2
1
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Counting Prime Numbers
How many primes exist which are less than 50?
1
1
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String of Exponents
Calculate the least integer greater than 5^{(\minus{}6)(\minus{}5)(\minus{}4)...(2)(3)(4)}.