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Princeton University Math Competition
2013 Princeton University Math Competition
2013 Princeton University Math Competition
Part of
Princeton University Math Competition
Subcontests
(16)
16
1
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2013 PUMaC Team 16
Is
cos
1
∘
\cos 1^\circ
cos
1
∘
rational? Prove.
15
1
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2013 PUMaC Team 15
Prove:
∣
sin
a
1
∣
+
∣
sin
a
2
∣
+
∣
sin
a
3
∣
+
…
+
∣
sin
a
n
∣
+
∣
cos
(
a
1
+
a
2
+
a
3
+
…
+
a
n
)
∣
≥
1.
|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.
∣
sin
a
1
∣
+
∣
sin
a
2
∣
+
∣
sin
a
3
∣
+
…
+
∣
sin
a
n
∣
+
∣
cos
(
a
1
+
a
2
+
a
3
+
…
+
a
n
)
∣
≥
1.
14
1
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2013 PUMaC Team 14
Shuffle a deck of
71
71
71
playing cards which contains
6
6
6
aces. Then turn up cards from the top until you see an ace. What is the average number of cards required to be turned up to find the first ace?
13
1
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2013 PUMaC Team 13
The equation
x
5
−
2
x
4
−
1
=
0
x^5-2x^4-1=0
x
5
−
2
x
4
−
1
=
0
has five complex roots
r
1
,
r
2
,
r
3
,
r
4
,
r
5
r_1,r_2,r_3,r_4,r_5
r
1
,
r
2
,
r
3
,
r
4
,
r
5
. Find the value of
1
r
1
8
+
1
r
2
8
+
1
r
3
8
+
1
r
4
8
+
1
r
5
8
.
\dfrac1{r_1^8}+\dfrac1{r_2^8}+\dfrac1{r_3^8}+\dfrac1{r_4^8}+\dfrac1{r_5^8}.
r
1
8
1
+
r
2
8
1
+
r
3
8
1
+
r
4
8
1
+
r
5
8
1
.
12
1
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2013 PUMaC Team 12
Let
D
D
D
be a point on the side
B
C
BC
BC
of
△
A
B
C
\triangle ABC
△
A
BC
. If
A
B
=
8
AB=8
A
B
=
8
,
A
C
=
7
AC=7
A
C
=
7
,
B
D
=
2
BD=2
B
D
=
2
, and
C
D
=
1
CD=1
C
D
=
1
, find
A
D
AD
A
D
.
11
1
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2013 PUMaC Team 11
If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?
10
1
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2013 PUMaC Team 10
On a plane, there are
7
7
7
seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else's). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?
9
1
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2013 PUMaC Team 9
If two distinct integers from
1
1
1
to
50
50
50
inclusive are chosen at random, what is the expected value of their product? Note: The expectation is defined as the sum of the products of probability and value, i.e., the expected value of a coin flip that gives you
$
10
\$10
$10
if head and
$
5
\$5
$5
if tail is
1
2
×
$
10
+
1
2
×
$
5
=
$
7.5
\tfrac12\times\$10+\tfrac12\times\$5=\$7.5
2
1
×
$10
+
2
1
×
$5
=
$7.5
.
8
7
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7
7
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6
6
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5
6
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4
7
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3
11
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2
10
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1
9
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