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Princeton University Math Competition
2008 Princeton University Math Competition
2008 Princeton University Math Competition
Part of
Princeton University Math Competition
Subcontests
(40)
1
1
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2008 PUMaC Team B1
Calculate
6
+
6
+
6
+
.
.
.
+
6
1
+
6
1
+
.
.
.
\sqrt{6 + \sqrt{6 + \sqrt{6 +... }}}+\frac{6}{1+ \frac{6}{1+...}}
6
+
6
+
6
+
...
+
1
+
1
+
...
6
6
10
1
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2008 PUMaC Team B10
Consider the sequence
s
0
=
(
1
,
2008
)
s_0 = (1, 2008)
s
0
=
(
1
,
2008
)
. Define new sequences
s
i
s_i
s
i
inductively by inserting the sum of each pair of adjacent terms in
s
i
−
1
s_{i-1}
s
i
−
1
between them — for instance,
s
1
=
(
1
,
2009
,
2008
)
s_1 = (1, 2009, 2008)
s
1
=
(
1
,
2009
,
2008
)
. For some
n
,
s
n
n, s_n
n
,
s
n
has exactly one term that appears twice. Find this repeated term.
9
1
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2008 PUMaC Team B9
Alex Lishkov is trying to guess sequence of
2009
2009
2009
random ternary digits (
0
,
1
0, 1
0
,
1
, or
2
2
2
). After he guesses each digit, he finds out whether he was right or not. If he guesses incorrectly, and
k
k
k
was the correct answer, then an oracle tells him what the next
k
k
k
digits will be. Being Bulgarian, Lishkov plays to maximize the expected number of digits guessed correctly. Let
P
n
P_n
P
n
be the probability that Lishkov guesses the nth digit correctly. Find
P
2009
P_{2009}
P
2009
. Write your answer in the form
x
+
y
R
e
(
ρ
k
)
x + yRe(\rho^k)
x
+
y
R
e
(
ρ
k
)
, where
x
x
x
and
y
y
y
are rational,
ρ
\rho
ρ
is complex, and
k
k
k
is a positive integer
8
1
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2008 PUMaC Team B8
Suppose that the roots of the quadratic
x
2
+
a
x
+
b
x^2 + ax + b
x
2
+
a
x
+
b
are
α
\alpha
α
and
β
\beta
β
. Then
α
3
\alpha^3
α
3
and
β
3
\beta^3
β
3
are the roots of some quadratic
x
2
+
c
x
+
d
x^2 + cx + d
x
2
+
c
x
+
d
. Find
c
c
c
in terms of
a
a
a
and
b
b
b
.
7
1
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2008 PUMaC Team B7
The graphs of the following equations divide the
x
y
xy
x
y
plane into some number of regions.
4
+
(
x
+
2
)
y
=
x
2
4 + (x + 2)y =x^2
4
+
(
x
+
2
)
y
=
x
2
(
x
+
2
)
2
+
y
2
=
16
(x + 2)^2 + y^2 =16
(
x
+
2
)
2
+
y
2
=
16
Find the area of the second smallest region.
4
1
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2008 PUMaC Team B4
What is the difference between the median and the mean of the following data set:
12
,
41
,
44
,
48
,
47
,
53
,
60
,
62
,
56
,
32
,
23
,
25
,
31
12,41, 44, 48, 47, 53, 60, 62, 56, 32, 23, 25, 31
12
,
41
,
44
,
48
,
47
,
53
,
60
,
62
,
56
,
32
,
23
,
25
,
31
?
6
1
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2008 PUMaC Team B6
The seven dwarves are at work on day when they find a large pile of diamonds. They want to split the diamonds evenly among them, but find that they would need to take away one diamond to split into seven equal piles. They are still arguing about this when they get home, so Snow White sends them to bed without supper. In the middle of the night, Sneezy wakes up and decides that he should get the extra diamond. So he puts one diamond aside, splits the remaining ones in to seven equal piles, and takes his pile along with the extra diamond. Then, he runs off with the diamonds. His sneeze wakes up Grumpy, who, thinking along the same lines, removes one diamond, divides the remainder into seven equal piles, and runs off. Finally, Sleepy, for the first time in his life, wakes up before sunrise and performs the same operation. When the remaining four dwarves arise, they find that the remaining diamonds can be split into
5
5
5
equal piles. Doc suggests that Snow White should get a share, so they have no problem splitting the remaining diamonds. Happy, Dopey, Bashful, Doc, and Snow White live happily ever after. What’s the smallest possible number of diamonds that the dwarves could have started out with?
3
1
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2008 PUMaC Team B3
What is the smallest positive integer value of
x
x
x
for which
x
≡
4
x \equiv 4
x
≡
4
(mod
9
9
9
) and
x
≡
7
x \equiv 7
x
≡
7
(mod
8
8
8
)?
2
1
Hide problems
2008 PUMaC Team B2
Find
log
2
3
∗
log
3
4
∗
log
4
5
∗
.
.
.
∗
log
62
63
∗
log
63
64
\log_2 3 * \log_3 4 * \log_4 5 * ... * \log_{62} 63 * \log_{63} 64
lo
g
2
3
∗
lo
g
3
4
∗
lo
g
4
5
∗
...
∗
lo
g
62
63
∗
lo
g
63
64
.
A1
1
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2008 PUMaC Individual Finals A1
Find all positive real numbers
b
b
b
for which there exists a positive real number
k
k
k
such that
n
−
k
≤
⌊
b
n
⌋
<
n
n-k \leq \left\lfloor bn \right\rfloor <n
n
−
k
≤
⌊
bn
⌋
<
n
for all positive integers
n
n
n
.
B7
1
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2008 PUMaC Number Theory B7
In this problem, we consider only polynomials with integer coeffients. Call two polynomials
p
p
p
and
q
q
q
really close if
p
(
2
k
+
1
)
≡
q
(
2
k
+
1
)
p(2k + 1) \equiv q(2k + 1)
p
(
2
k
+
1
)
≡
q
(
2
k
+
1
)
(mod
210
210
210
) for all
k
∈
Z
+
k \in Z^+
k
∈
Z
+
. Call a polynomial
p
p
p
partial credit if no polynomial of lesser degree is really close to it. What is the maximum possible degree of partial credit?
A10/B10
1
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2008 PUMaC Number Theory A10 / B10
What is the smallest number
n
n
n
such that you can choose
n
n
n
distinct odd integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,..., a_n
a
1
,
a
2
,
...
,
a
n
, none of them
1
1
1
, with
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
=
1
\frac{1}{a_1}+ \frac{1}{a_2}+ ...+ \frac{1}{a_n}= 1
a
1
1
+
a
2
1
+
...
+
a
n
1
=
1
?
A5
1
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2008 PUMaC Number Theory A5
If
f
(
x
)
=
x
x
x
x
f(x) = x^{x^{x^x}}
f
(
x
)
=
x
x
x
x
, find the last two digits of
f
(
17
)
+
f
(
18
)
+
f
(
19
)
+
f
(
20
)
f(17) + f(18) + f(19) + f(20)
f
(
17
)
+
f
(
18
)
+
f
(
19
)
+
f
(
20
)
.
A3/B4
1
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2008 PUMaC Number Theory A3 / B4
Find the largest integer
n
n
n
, where
200
9
n
2009^n
200
9
n
divides
200
8
200
9
2010
+
201
0
200
9
2008
2008^{2009^{2010}} + 2010^{2009^{2008}}
200
8
200
9
2010
+
201
0
200
9
2008
.
A6/B8
2
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2008 PUMaC Combinatorics A6 / B8
x
x
x
x
xxxx
xxxx
x
x
xx
xx
x
x
x
x
x
x
In how many ways can you fill in the
x
x
x
s with the numbers
1
−
8
1-8
1
−
8
so that for each
x
x
x
, the numbers below and to the right are higher.
2008 PUMaC Number Theory A6 / B8
What is the largest integer which cannot be expressed as
2008
x
+
2009
y
+
2010
z
2008x + 2009y + 2010z
2008
x
+
2009
y
+
2010
z
for some positive integers
x
,
y
x, y
x
,
y
, and
z
z
z
?
A4/B6
2
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2008 PUMaC Combinatorics A4 / B6
Find the sum of the values of
x
x
x
for which
(
x
0
)
−
(
x
1
)
+
(
x
2
)
−
.
.
.
+
(
x
2008
)
=
0
\binom{x}{0}-\binom{x}{1}+\binom{x}{2}-...+\binom{x}{2008}=0
(
0
x
)
−
(
1
x
)
+
(
2
x
)
−
...
+
(
2008
x
)
=
0
2008 PUMaC Number Theory A4 / B6
f
(
n
)
f(n)
f
(
n
)
is the sum of all integers less than
n
n
n
and relatively prime to
n
n
n
. Find all integers
n
n
n
such that there exist integers
k
k
k
and
ℓ
\ell
ℓ
such that
f
(
n
k
)
=
n
ℓ
f(n^k) = n^{\ell}
f
(
n
k
)
=
n
ℓ
.
A3/B5
1
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2008 PUMaC Combinatorics A3 / B5
Evaluate
∑
m
=
0
2009
∑
n
=
0
m
(
2009
m
)
(
m
n
)
\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}
∑
m
=
0
2009
∑
n
=
0
m
(
m
2009
)
(
n
m
)
A5/B7
1
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2008 PUMaC Combinatorics A5 / B7
In how many ways can Alice, Bob, Charlie, David, and Eve split
18
18
18
marbles among themselves so that no two of them have the same number of marbles?
B2
2
Hide problems
2008 PUMaC Algebra B2
What is
3
(
2
log
4
(
2
(
2
log
3
9
)
)
)
3(2 \log_4 (2(2 \log_3 9)))
3
(
2
lo
g
4
(
2
(
2
lo
g
3
9
)))
?
2008 PUMaC Individual Finals B2
Let
P
P
P
be a convex polygon, and let
n
≥
3
n \ge 3
n
≥
3
be a positive integer. On each side of
P
P
P
, erect a regular
n
n
n
-gon that shares that side of
P
P
P
, and is outside
P
P
P
. If none of the interiors of these regular n-gons overlap, we call P
n
n
n
-good. (a) Find the largest value of
n
n
n
such that every convex polygon is
n
n
n
-good. (b) Find the smallest value of
n
n
n
such that no convex polygon is
n
n
n
-good.
A9/B10
2
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2008 PUMaC Algebra A9 / B10
If
p
(
x
)
p(x)
p
(
x
)
is a polynomial with integer coeffcients, let
q
(
x
)
=
p
(
x
)
x
(
1
−
x
)
q(x) = \frac{p(x)}{x(1-x)}
q
(
x
)
=
x
(
1
−
x
)
p
(
x
)
. If
q
(
x
)
=
q
(
1
1
−
x
)
q(x) = q\left(\frac{1}{1-x}\right)
q
(
x
)
=
q
(
1
−
x
1
)
for every
x
≠
0
x \ne 0
x
=
0
, and
p
(
2
)
=
−
7
,
p
(
3
)
=
−
11
p(2) = -7, p(3) = -11
p
(
2
)
=
−
7
,
p
(
3
)
=
−
11
, find
p
(
10
)
p(10)
p
(
10
)
.
2008 PUMaC Combinatorics A9 / B10
How many spanning trees does the following graph (with
6
6
6
vertices and
9
9
9
edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png
A8/B9
3
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2008 PUMaC Algebra A8 / B9
Find the polynomial
f
f
f
with the following properties:
∙
\bullet
∙
its leading coefficient is
1
1
1
,
∙
\bullet
∙
its coefficients are nonnegative integers,
∙
\bullet
∙
72
∣
f
(
x
)
72|f(x)
72∣
f
(
x
)
if
x
x
x
is an integer,
∙
\bullet
∙
if
g
g
g
is another polynomial with the same properties, then
g
−
f
g - f
g
−
f
has a nonnegative leading coecient.
2008 PUMaC Combinatorics A8 / B9
A SET cards have four characteristics: number, color, shape, and shading, each of which has
3
3
3
values. A SET deck has
81
81
81
cards, one for each combination of these values. A SET is three cards such that, for each characteristic, the values of the three cards for that characteristics are either all the same or all different. In how many ways can you replace each SET card in the deck with another SET card (possibly the same), with no card used twice, such that any three cards that were a SET before are still a SET? (Alternately, a SET card is an ordered
4
4
4
-tuple of
0
0
0
s,
1
1
1
s, and
2
2
2
s, and three cards form a SET if their sum is (
0
,
0
,
0
,
0
0, 0, 0, 0
0
,
0
,
0
,
0
) mod
3
3
3
, for instance, (
0
,
1
,
2
,
2
0, 1, 2, 2
0
,
1
,
2
,
2
), (
1
,
0
,
2
,
1
1, 0, 2, 1
1
,
0
,
2
,
1
), and (
2
,
2
,
2
,
0
2, 2, 2, 0
2
,
2
,
2
,
0
) form a SET. How many permutations of the SET cards maintain SET-ness?)
2008 PUMaC Number Theory A8 / B9
Find all sets of three primes
p
,
q
p, q
p
,
q
, and
r
r
r
such that
p
+
q
=
r
p + q = r
p
+
q
=
r
and
(
r
−
p
)
(
q
−
p
)
−
27
p
(r -p)(q - p) - 27p
(
r
−
p
)
(
q
−
p
)
−
27
p
is a perfect square.
A7
3
Hide problems
2008 PUMaC Algebra A7
Suppose
x
9
=
1
x^9 = 1
x
9
=
1
but
x
3
≠
1
x^3 \ne 1
x
3
=
1
. Find a polynomial of minimal degree equal to
1
1
+
x
\frac{1}{1+x}
1
+
x
1
.
2008 PUMaC Combinatorics A7
Joe makes two cubes of sidelengths
9
9
9
and
10
10
10
from
1729
1729
1729
randomly oriented and randomly arranged unit cubes, which are initially unpainted. These cubes are dipped into white paint. Then two cubes of sidelengths
1
1
1
and
12
12
12
are formed from the same unit cubes, again randomly oriented and randomly arranged, and these cubes are dipped into paint remover. Joe continues to alternately dip cubes of sides
9
9
9
and
10
10
10
into paint and cubes of sides
1
1
1
and
12
12
12
into paint remover ad nauseam. What is the limit of the expected number of painted unit cube faces immediately after dipping in paint remover?
2008 PUMaC Number Theory A7
Find the smallest positive integer
n
n
n
such that
3
2
n
=
167
x
+
2
32^n = 167x + 2
3
2
n
=
167
x
+
2
for some integer
x
x
x
A3/B6
1
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2008 PUMaC Algebra A3 / B6
Let
f
(
n
)
=
9
n
5
−
5
n
3
−
4
n
f(n) = 9n^5- 5n^3 - 4n
f
(
n
)
=
9
n
5
−
5
n
3
−
4
n
. Find the greatest common divisor of
f
(
17
)
,
f
(
18
)
,
.
.
.
,
f
(
2009
)
f(17), f(18),... ,f(2009)
f
(
17
)
,
f
(
18
)
,
...
,
f
(
2009
)
.
A2
1
Hide problems
2008 PUMaC Algebra A2
What is the polynomial of smallest degree that passes through
(
−
2
,
2
)
,
(
−
1
,
1
)
,
(
0
,
2
)
,
(
1
,
−
1
)
(-2, 2), (-1, 1), (0, 2),(1,-1)
(
−
2
,
2
)
,
(
−
1
,
1
)
,
(
0
,
2
)
,
(
1
,
−
1
)
, and
(
2
,
10
)
(2, 10)
(
2
,
10
)
?
A1/B3
1
Hide problems
2008 PUMaC Algebra A1 / B3
Given the sequence
1
,
2
,
1
,
2
,
2
,
1
,
2
,
2
,
2
,
1
,
2
,
2
,
2
,
2
,
1
,
.
.
.
,
1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,
1
,
2
,
1
,
2
,
2
,
1
,
2
,
2
,
2
,
1
,
2
,
2
,
2
,
2
,
1
,
...
,
find
n
n
n
such that the sum of the first
n
n
n
terms is
2008
2008
2008
or
2009
2009
2009
.
5
1
Hide problems
2008 PUMaC Team B5
Quadrilateral
A
B
C
D
ABCD
A
BC
D
has both an inscribed and a circumscribed circle and sidelengths
B
C
=
4
,
C
D
=
5
,
D
A
=
6
BC = 4, CD = 5, DA = 6
BC
=
4
,
C
D
=
5
,
D
A
=
6
. Find the area of
A
B
C
D
ABCD
A
BC
D
.
B6
1
Hide problems
2008 PUMaC Geometry B6
Circles
A
,
B
A, B
A
,
B
, and
C
C
C
each have radius
r
r
r
, and their centers are the vertices of an equilateral triangle of side length
6
r
6r
6
r
. Two lines are drawn, one tangent to
A
A
A
and
C
C
C
and one tangent to
B
B
B
and
C
C
C
, such that
A
A
A
is on the opposite side of each line from
B
B
B
and
C
C
C
. Find the sine of the angle between the two lines.http://4.bp.blogspot.com/-IZv8q-3NYZg/XXmrroy2PnI/AAAAAAAAKxg/jSOcOOQ8Kyw0EwHUifXJ1jOd2ENAo1FfACK4BGAYYCw/s200/2008%2Bpumac%2Bb6.png
B5
3
Hide problems
2008 PUMaC Geometry B5
Two externally tangent circles have radius
2
2
2
and radius
3
3
3
. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?
2008 PUMaC Algebra B5
How many real roots do
x
5
+
3
x
4
−
4
x
3
−
8
x
2
+
6
x
−
1
x^5 +3x^4 -4x^3 -8x^2 +6x-1
x
5
+
3
x
4
−
4
x
3
−
8
x
2
+
6
x
−
1
and
x
5
−
3
x
4
−
2
x
3
−
3
x
2
−
6
x
+
1
x^5-3x^4 -2x^3 -3x^2 -6x+1
x
5
−
3
x
4
−
2
x
3
−
3
x
2
−
6
x
+
1
share?
2008 PUMaC Number Theory B5
How many integers
n
n
n
are there such that
0
≤
n
≤
720
0 \le n \le 720
0
≤
n
≤
720
and
n
2
≡
1
n^2 \equiv 1
n
2
≡
1
(mod
720
720
720
)?
B4
3
Hide problems
2008 PUMaC Geometry B4
A cube is divided into
27
27
27
unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these
8
8
8
spheres. What is the smallest possible value for the radius of the last sphere?
2008 PUMaC Combinatorics B4
A
2008
×
2009
2008 \times 2009
2008
×
2009
rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with
1
×
2
1 \times 2
1
×
2
dominoes?
2008 PUMaC Algebra B4
Find the product of the minimum and maximum values of
3
x
+
1
9
x
2
+
6
x
+
2
\frac{3x+1}{9x^2+6x+2}
9
x
2
+
6
x
+
2
3
x
+
1
.
B1
5
Show problems
A10
3
Hide problems
2008 PUMaC Geometry A10/B10
A cuboctahedron is the convex hull of (smallest convex set containing) the
12
12
12
points
(
±
1
,
±
1
,
0
)
,
(
±
1
,
0
,
±
1
)
,
(
0
,
±
1
,
±
1
)
(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)
(
±
1
,
±
1
,
0
)
,
(
±
1
,
0
,
±
1
)
,
(
0
,
±
1
,
±
1
)
. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the fi gure. The area of that set is the solid angle of the fi gure as viewed from the origin.)
2008 PUMaC Algebra A10
Find the sum of all integer values of
n
n
n
such that the equation
x
(
y
z
)
2
+
y
(
z
x
)
2
+
z
(
x
y
)
2
=
n
\frac{x}{(yz)^2} + \frac{y}{(zx)^2} + \frac{z}{(xy)^2} = n
(
yz
)
2
x
+
(
z
x
)
2
y
+
(
x
y
)
2
z
=
n
has a solution in positive integers.
2008 PUMaC Combinatorics A10
In his youth, Professor John Horton Conway lived on a farm with
2009
2009
2009
cows. Conway wishes to move the cows from the negative
x
x
x
axis to the positive
x
x
x
axis. The cows are initially lined up in order
1
,
2
,
.
.
.
,
2009
1, 2, . . . , 2009
1
,
2
,
...
,
2009
on the negative
x
x
x
axis. Conway can give two possible commands to the cows. One is the PUSH command, upon which the first cow from the negative
x
x
x
axis moves to the lowest position on the positive
y
y
y
axis. The other is the POP command, upon which the cow in the lowest position on the
y
y
y
axis moves to the positive
x
x
x
axis. For example, if Conway says PUSH POP
2009
2009
2009
times, then the resulting permutation of cows is the same,
1
,
2
,
.
.
.
,
2009
1, 2, . . . , 2009
1
,
2
,
...
,
2009
. If Conway says PUSH
2009
2009
2009
times followed by POP
2009
2009
2009
times, the resulting permutation of cows is
2009
,
.
.
.
,
2
,
1
2009, . . . , 2, 1
2009
,
...
,
2
,
1
. How many output permutations are possible after Conway finishes moving all the cows from the negative
x
x
x
axis to the positive
x
x
x
axis?
A9
2
Hide problems
2008 PUMaC Geometry A9
In tetrahedron
A
B
C
D
ABCD
A
BC
D
with circumradius
2
2
2
,
A
B
=
2
AB = 2
A
B
=
2
,
C
D
=
7
CD = \sqrt{7}
C
D
=
7
, and
∠
A
B
C
=
∠
B
A
D
=
π
2
\angle ABC = \angle BAD = \frac{\pi}{2}
∠
A
BC
=
∠
B
A
D
=
2
π
. Find all possible angles between the planes containing
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
.
2008 PUMaC Number Theory A9
Find the number of positive integer solutions of
(
x
2
+
2
)
(
y
2
+
3
)
(
z
2
+
4
)
=
60
x
y
z
(x^2 + 2)(y^2 + 3)(z^2 + 4) = 60xyz
(
x
2
+
2
)
(
y
2
+
3
)
(
z
2
+
4
)
=
60
x
yz
.
A8
1
Hide problems
2008 PUMaC Geometry A8
In four-dimensional space, the
24
24
24
-cell of sidelength
2
\sqrt{2}
2
is the convex hull of (smallest convex set containing) the
24
24
24
points
(
±
1
,
±
1
,
0
,
0
)
(\pm 1, \pm 1, 0, 0)
(
±
1
,
±
1
,
0
,
0
)
and its permutations. Find the four-dimensional volume of this region.
A7/B9
1
Hide problems
2008 PUMaC Geometry A7/B9
Let
H
\mathcal{H}
H
be the region of points
(
x
,
y
)
(x, y)
(
x
,
y
)
, such that
(
1
,
0
)
,
(
x
,
y
)
,
(
−
x
,
y
)
(1, 0), (x, y), (-x, y)
(
1
,
0
)
,
(
x
,
y
)
,
(
−
x
,
y
)
, and
(
−
1
,
0
)
(-1,0)
(
−
1
,
0
)
form an isosceles trapezoid whose legs are shorter than the base between
(
x
,
y
)
(x, y)
(
x
,
y
)
and
(
−
x
,
y
)
(-x,y)
(
−
x
,
y
)
. Find the least possible positive slope that a line could have without intersecting
H
\mathcal{H}
H
.
A6
2
Hide problems
2008 PUMaC Geometry A6
Find the coordinates of the point in the plane at which the sum of the distances from it to the three points
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
2
,
0
)
(2, 0)
(
2
,
0
)
,
(
0
,
3
)
(0, \sqrt{3})
(
0
,
3
)
is minimal.
2008 PUMaC Algebra A6
Let
x
x
x
be the largest root of
x
4
−
2009
x
+
1
x^4 - 2009x + 1
x
4
−
2009
x
+
1
. Find the nearest integer to
1
x
3
−
2009
\frac{1}{x^3-2009}
x
3
−
2009
1
.
A5/B8
2
Hide problems
2008 PUMaC Geometry A5/B8
Infinitesimal Randall Munroe is glued to the center of a pentagon with side length
1
1
1
. At each corner of the pentagon is a confused infinitesimal velociraptor. At any time, each raptor is running at one unit per second directly towards the next raptor in the pentagon (in counterclockwise order). How far does each confused raptor travel before it reaches Randall Munroe?
2008 PUMaC Algebra A5 / B8
Let
H
k
=
Σ
i
=
1
k
1
i
H_k =\Sigma_{i=1}^k \frac{1}{i}
H
k
=
Σ
i
=
1
k
i
1
for all positive integers
k
k
k
. Find an closed-form expression for
Σ
i
=
1
k
H
i
\Sigma_{i=1}^k H_i
Σ
i
=
1
k
H
i
in terms of
n
n
n
and
H
n
H_n
H
n
.
A4/B7
2
Hide problems
2008 PUMaC Geometry A4/B7
How many ordered pairs of real numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
are there such that
x
2
+
y
2
=
200
x^2+y^2 = 200
x
2
+
y
2
=
200
and
(
x
−
5
)
2
+
(
y
−
5
)
2
+
(
x
+
5
)
2
+
(
y
+
5
)
2
\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}
(
x
−
5
)
2
+
(
y
−
5
)
2
+
(
x
+
5
)
2
+
(
y
+
5
)
2
is an integer?
2008 PUMaC Algebra A4 / B7
What's the greatest integer
n
n
n
for which the system
k
<
x
k
<
k
+
1
k < x^k < k + 1
k
<
x
k
<
k
+
1
for
k
=
1
,
2
,
.
.
.
,
n
k = 1,2,..., n
k
=
1
,
2
,
...
,
n
has a solution?
A3
2
Hide problems
2008 PUMaC Geometry A3
Consider a
12
12
12
-sided regular polygon. If the vertices going clockwise are
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
,
F
F
F
, etc, draw a line between
A
A
A
and
F
F
F
,
B
B
B
and
G
G
G
,
C
C
C
and
H
H
H
, etc. This will form a smaller
12
12
12
-sided regular polygon in the center of the larger one. What is the area of the smaller one divided by the area of the larger one?
2008 PUMaC Individual Finals A3
A sequence
{
a
i
}
\{a_i\}
{
a
i
}
is defined by
a
1
=
c
a_1 = c
a
1
=
c
for some
c
>
0
c > 0
c
>
0
and
a
n
+
1
=
a
n
+
n
a
n
a_{n+1} = a_n + \frac{n}{a_n}
a
n
+
1
=
a
n
+
a
n
n
. Prove that
a
n
n
\frac{a_n}{n}
n
a
n
converges and find its limit.
A2/B3
4
Show problems
A1/B2
3
Hide problems
2008 PUMaC Geometry A1/B2
What is the area of a circle with a circumference of
8
8
8
?
2008 PUMaC Number Theory A1 / B2
How many zeros are there at the end of
792
!
792!
792
!
when written in base
10
10
10
?
2008 PUMaC Combinatorics A1 / B2
How many
3
3
3
-digit numbers contain the digit
7
7
7
exactly once?