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Princeton University Math Competition
2012 Princeton University Math Competition
2012 Princeton University Math Competition
Part of
Princeton University Math Competition
Subcontests
(27)
Team Round
1
Hide problems
Pumac 2012 Team Round
Time limit: 20 minutes.Fill in the crossword above with answers to the problems below.Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1,"
e
4
π
i
/
3
e^{4\pi i/3}
e
4
πi
/3
, and
e
5
π
i
/
3
e^{5\pi i/3}
e
5
πi
/3
. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across",
e
4
π
i
/
3
e^{4\pi i/3}
e
4
πi
/3
means "down and to the left," and
e
5
π
i
/
3
e^{5\pi i/3}
e
5
πi
/3
means "down and to the right."To fill in the answer to, for example,
12
12
12
across, start at the hexagon labeled
12
12
12
, and write the digits, proceeding to the right along the gray line. (Note:
12
12
12
across has space for exactly
5
5
5
digits.)Each hexagon is worth one point, and must be filled by something from the set
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
. Note that
π
\pi
π
is not in the set, and neither is
i
i
i
, nor
2
\sqrt2
2
, nor
♡
\heartsuit
♡
,etc.None of the answers will begin with a
0
0
0
."Concatenate
a
a
a
and
b
b
b
" means to write the digits of
a
a
a
, followed by the digits of
b
b
b
. For example, concatenating
10
10
10
and
3
3
3
gives
103
103
103
. (It's not the same as concatenating
3
3
3
and
10
10
10
.)Calculators are allowed!THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT.Good luck, and have fun! https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.pngAcross (1) A 3. (3 digits) Suppose you draw
5
5
5
vertices of a convex pentagon (but not the sides!). Let
N
N
N
be the number of ways you can draw at least
0
0
0
straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is
N
−
64
N - 64
N
−
64
? (Note what the question is asking for! You have been warned!)A 5. (3 digits) Among integers
{
1
,
2
,
.
.
.
,
1
0
2012
}
\{1, 2,..., 10^{2012}\}
{
1
,
2
,
...
,
1
0
2012
}
, let
n
n
n
be the number of numbers for which the sum of the digits is divisible by
5
5
5
. What are the first three digits (from the left) of
n
n
n
?A 6. (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly
2012
2012
2012
digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If
n
2
n^2
n
2
is the largest perfect square he writes, what is
n
n
n
?A 8. (3 digits) How many positive integers
n
n
n
are there such that
n
≤
2012
n \le 2012
n
≤
2012
, and the greatest common divisor of
n
n
n
and
2012
2012
2012
is a prime number?A 9. (4 digits) I have a random number machine generator that is very good at generating integers between
1
1
1
and
256
256
256
, inclusive, with equal probability. However, right now, I want to produce a random number between
1
1
1
and
n
n
n
, inclusive, so I do the following:
∙
\bullet
∙
I use my machine to generate a number between
1
1
1
and
256
256
256
. Call this
a
a
a
.
∙
\bullet
∙
I take a and divide it by
n
n
n
to get remainder
r
r
r
. If
r
≠
0
r \ne 0
r
=
0
, then I record
r
r
r
as the randomly generated number. If
r
=
0
r = 0
r
=
0
, then I record
n
n
n
instead. Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between
1
1
1
and
10
10
10
. Let
p
p
p
be the probability that the two numbers are equal. What is
p
⋅
2
16
p \cdot 2^{16}
p
⋅
2
16
?A 12. (5 digits) You and your friend play the following dangerous game. You two start off at some point
(
x
,
y
)
(x, y)
(
x
,
y
)
on the plane, where
x
x
x
and
y
y
y
are nonnegative integers. When it is player
A
A
A
's turn, A tells his opponent
B
B
B
to move to another point on the plane. Then
A
A
A
waits for a while. If
B
B
B
is not eaten by a tiger, then
A
A
A
moves to that point as well. From a point
(
x
,
y
)
(x, y)
(
x
,
y
)
there are three places
A
A
A
can tell
B
B
B
to walk to: leftwards to
(
x
−
1
,
y
)
(x - 1, y)
(
x
−
1
,
y
)
, downwards to
(
x
,
y
−
1
)
(x, y-1)
(
x
,
y
−
1
)
, and simultaneously downwards and leftwards to
(
x
−
1
,
y
−
1
)
(x-1, y-1)
(
x
−
1
,
y
−
1
)
. However, you cannot move to a point with a negative coordinate. Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten. Consider all possible starting points
(
x
,
y
)
(x, y)
(
x
,
y
)
with
0
≤
x
≤
346
0 \le x \le 346
0
≤
x
≤
346
and
0
≤
y
≤
346
0 \le y \le 346
0
≤
y
≤
346
, and
x
x
x
and
y
y
y
are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win?Down and to the left
e
4
π
i
/
3
e^{4\pi i/3}
e
4
πi
/3
DL 2. (2 digits) ABCDE is a pentagon with
A
B
=
B
C
=
C
D
=
2
AB = BC = CD = \sqrt2
A
B
=
BC
=
C
D
=
2
,
∠
A
B
C
=
∠
B
C
D
=
120
\angle ABC = \angle BCD = 120
∠
A
BC
=
∠
BC
D
=
120
degrees, and
∠
B
A
E
=
∠
C
D
E
=
105
\angle BAE = \angle CDE = 105
∠
B
A
E
=
∠
C
D
E
=
105
degrees. Find the area of triangle
△
B
D
E
\vartriangle BDE
△
B
D
E
. Your answer in its simplest form can be written as
a
+
b
c
\frac{a+\sqrt{b}}{c}
c
a
+
b
, where where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers and
b
b
b
is square-free. Find
a
b
c
abc
ab
c
.DL 3. (3 digits) Suppose
x
x
x
and
y
y
y
are integers which satisfy
4
x
2
y
2
+
25
y
2
x
2
=
10055
x
2
+
4022
y
2
+
2012
x
2
y
2
−
20.
\frac{4x^2}{y^2} + \frac{25y^2}{x^2} = \frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20.
y
2
4
x
2
+
x
2
25
y
2
=
x
2
10055
+
y
2
4022
+
x
2
y
2
2012
−
20.
What is the maximum possible value of
x
y
−
1
xy -1
x
y
−
1
?DL 5. (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties:
∙
\bullet
∙
The square has side length
7
2
7\sqrt2
7
2
.
∙
\bullet
∙
The boundary of the square intersects the graph of
x
y
=
0
xy = 0
x
y
=
0
at at least
3
3
3
points.DL 8. (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred:
∙
\bullet
∙
Mom: Tell me the number of zeros at the end of
2012
!
2012!
2012
!
∙
\bullet
∙
PT: Huh?
2012
2012
2012
ends in
2
2
2
, so there aren't any zeros.
∙
\bullet
∙
Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about
2012
2012
2012
, I was asking about
2012
!
2012!
2012
!
. What is the correct answer?DL 9. (4 digits) Define the following:
∙
\bullet
∙
A
=
∑
n
=
1
∞
1
n
6
A = \sum^{\infty}_{n=1}\frac{1}{n^6}
A
=
∑
n
=
1
∞
n
6
1
∙
\bullet
∙
B
=
∑
n
=
1
∞
1
n
6
+
1
B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}
B
=
∑
n
=
1
∞
n
6
+
1
1
∙
\bullet
∙
C
=
∑
n
=
1
∞
1
(
n
+
1
)
6
C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}
C
=
∑
n
=
1
∞
(
n
+
1
)
6
1
∙
\bullet
∙
D
=
∑
n
=
1
∞
1
(
2
n
−
1
)
6
D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}
D
=
∑
n
=
1
∞
(
2
n
−
1
)
6
1
∙
\bullet
∙
E
=
∑
n
=
1
∞
1
(
2
n
+
1
)
6
E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}
E
=
∑
n
=
1
∞
(
2
n
+
1
)
6
1
Consider the ratios
B
A
,
C
A
,
D
A
,
E
A
\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}
A
B
,
A
C
,
A
D
,
A
E
. Exactly one of the four is a rational number. Let that number be
r
/
s
r/s
r
/
s
, where
r
r
r
and
s
s
s
are nonnegative integers and
g
c
d
(
r
,
s
)
=
1
gcd \,(r, s) = 1
g
c
d
(
r
,
s
)
=
1
. Concatenate
r
,
s
r, s
r
,
s
. (It might be helpful to know that
A
=
π
6
945
A = \frac{\pi^6}{945}
A
=
945
π
6
.)DL 10. (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at
(
−
1
,
0
)
(-1, 0)
(
−
1
,
0
)
,
(
1
,
0
)
(1, 0)
(
1
,
0
)
,
(
1
,
100
)
(1, 100)
(
1
,
100
)
,
(
−
1
,
100
)
(-1, 100)
(
−
1
,
100
)
. You remove a section of the bottom of the paper by cutting along the function
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
, where
f
f
f
satisfies
f
(
1
)
=
f
(
−
1
)
=
0
f(1) = f(-1) = 0
f
(
1
)
=
f
(
−
1
)
=
0
. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an
L
L
L
-shaped elbow tube. We can write
f
(
1
3
)
+
f
(
1
6
)
=
a
+
b
π
c
f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}
f
(
3
1
)
+
f
(
6
1
)
=
π
c
a
+
b
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers and
b
b
b
is square-free.
F
i
n
d
a
+
b
+
c
Find a+b+c
F
in
d
a
+
b
+
c
.DL 11. (3 digits) Let
Ξ
(
x
)
=
2012
(
x
−
2
)
2
+
278
(
x
−
2
)
2012
+
e
x
2
−
4
x
+
4
+
1392
+
(
x
2
−
4
x
+
4
)
e
x
2
−
4
x
+
4
\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}
Ξ
(
x
)
=
2012
(
x
−
2
)
2
+
278
(
x
−
2
)
2012
+
e
x
2
−
4
x
+
4
+
1392
+
(
x
2
−
4
x
+
4
)
e
x
2
−
4
x
+
4
find the area of the region in the
x
y
xy
x
y
-plane satisfying:
{
x
≥
0
a
n
d
x
≤
4
a
n
d
y
≥
0
a
n
d
y
≤
Ξ
(
x
)
}
\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}
{
x
≥
0
an
d
x
≤
4
an
d
y
≥
0
an
d
y
≤
Ξ
(
x
)
}
DL 13. (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of
2
2
2
, and the other cone has a radius of
3
3
3
. The two cones with radii
2
2
2
have height
4
4
4
, and the other cone has height
6
6
6
. Let
V
V
V
be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height
6
6
6
. Find
V
2
V^2
V
2
.Down and to the right
e
5
π
i
/
3
e^{5\pi i/3}
e
5
πi
/3
DR 1. (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as
r
/
s
r/s
r
/
s
, where r and s are integers and
g
c
d
(
r
,
s
)
=
1
gcd \,(r, s) = 1
g
c
d
(
r
,
s
)
=
1
. What is
r
+
s
r + s
r
+
s
?DR 4. (3 digits) Let
a
1
=
2
+
2
a_1 = 2 +\sqrt2
a
1
=
2
+
2
and
b
1
=
2
b_1 =\sqrt2
b
1
=
2
, and for
n
≥
1
n \ge 1
n
≥
1
,
a
n
+
1
=
∣
a
n
−
b
n
∣
a_{n+1} = |a_n - b_n|
a
n
+
1
=
∣
a
n
−
b
n
∣
and
b
n
+
1
=
a
n
+
b
n
b_{n+1} = a_n + b_n
b
n
+
1
=
a
n
+
b
n
. The minimum value of
a
n
2
+
a
n
b
n
−
6
b
n
2
6
b
n
2
−
a
n
2
\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}
6
b
n
2
−
a
n
2
a
n
2
+
a
n
b
n
−
6
b
n
2
can be written in the form
a
b
−
c
a\sqrt{b} - c
a
b
−
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers and
b
b
b
is square-free. Concatenate
c
,
b
,
a
c, b, a
c
,
b
,
a
(in that order!).DR 7. (3 digits) How many solutions are there to
a
503
+
b
1006
=
c
2012
a^{503} + b^{1006} = c^{2012}
a
503
+
b
1006
=
c
2012
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are integers and
∣
a
∣
|a|
∣
a
∣
,
∣
b
∣
|b|
∣
b
∣
,
∣
c
∣
|c|
∣
c
∣
are all less than
2012
2012
2012
? PS. You should use hide for answers.
B6
1
Hide problems
2012 PUMaC Number Theory B6
Let
f
n
(
x
)
=
n
+
x
2
f_n(x) = n + x^2
f
n
(
x
)
=
n
+
x
2
. Evaluate the product
g
c
d
{
f
2001
(
2002
)
,
f
2001
(
2003
)
}
×
g
c
d
{
f
2011
(
2012
)
,
f
2011
(
2013
)
}
×
g
c
d
{
f
2021
(
2022
)
,
f
2021
(
2023
)
}
gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\} \times gcd\{f_{2021}(2022), f_{2021}(2023)\}
g
c
d
{
f
2001
(
2002
)
,
f
2001
(
2003
)}
×
g
c
d
{
f
2011
(
2012
)
,
f
2011
(
2013
)}
×
g
c
d
{
f
2021
(
2022
)
,
f
2021
(
2023
)}
, where
g
c
d
{
x
,
y
}
gcd\{x, y\}
g
c
d
{
x
,
y
}
is the greatest common divisor of
x
x
x
and
y
y
y
B5
1
Hide problems
2012 PUMaC Algebra B5
Considering all numbers of the form
n
=
⌊
k
3
2012
⌋
n = \lfloor \frac{k^3}{2012} \rfloor
n
=
⌊
2012
k
3
⌋
, where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
, and
k
k
k
ranges from
1
1
1
to
2012
2012
2012
, how many of these
n
n
n
’s are distinct?
A4 / B7
2
Hide problems
2012 PUMaC Algebra A4 / B7
Let f be a polynomial of degree
3
3
3
with integer coefficients such that
f
(
0
)
=
3
f(0) = 3
f
(
0
)
=
3
and
f
(
1
)
=
11
f(1) = 11
f
(
1
)
=
11
. If f has exactly
2
2
2
integer roots, how many such polynomials
f
f
f
exist?
2012 PUMaC Number Theory A4 / B7
Find the sum of all possible sums
a
+
b
a + b
a
+
b
where
a
a
a
and
b
b
b
are nonnegative integers such that
4
a
+
2
b
+
5
4^a + 2^b + 5
4
a
+
2
b
+
5
is a perfect square.
A3 / B6
1
Hide problems
2012 PUMaC Algebra A3 / B6
Compute
Σ
n
=
1
∞
n
+
1
n
2
(
n
+
2
)
2
\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}
Σ
n
=
1
∞
n
2
(
n
+
2
)
2
n
+
1
. Your answer in simplest form can be written as
a
/
b
a/b
a
/
b
, where
a
,
b
a, b
a
,
b
are relatively-prime positive integers. Find
a
+
b
a + b
a
+
b
.
A2 / B4
1
Hide problems
2012 PUMaC Algebra A2 / B4
If
x
,
y
x, y
x
,
y
, and
z
z
z
are real numbers with
x
−
y
z
+
y
−
z
x
+
z
−
x
y
=
36
\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36
z
x
−
y
+
x
y
−
z
+
y
z
−
x
=
36
, find
2012
+
x
−
y
z
⋅
y
−
z
x
⋅
z
−
x
y
2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}
2012
+
z
x
−
y
⋅
x
y
−
z
⋅
y
z
−
x
.
B4
1
Hide problems
2012 PUMaC Combinatorics B4
For a set
S
S
S
of integers, define
max
(
S
)
\max (S)
max
(
S
)
to be the maximal element of
S
S
S
. How many non-empty subsets
S
⊆
{
1
,
2
,
3
,
.
.
.
,
10
}
S \subseteq \{1, 2, 3, ... , 10\}
S
⊆
{
1
,
2
,
3
,
...
,
10
}
satisfy
max
(
S
)
≤
∣
S
∣
+
2
\max (S) \le |S| + 2
max
(
S
)
≤
∣
S
∣
+
2
?
A7 / B8
2
Hide problems
2012 PUMaC Combinatorics A7 / B8
A PUMaC grader is grading the submissions of forty students
s
1
,
s
2
,
.
.
.
,
s
40
s_1, s_2, ..., s_{40}
s
1
,
s
2
,
...
,
s
40
for the individual finals round, which has three problems. After grading a problem of student
s
i
s_i
s
i
, the grader either:
∙
\bullet
∙
grades another problem of the same student, or
∙
\bullet
∙
grades the same problem of the student
s
i
−
1
s_{i-1}
s
i
−
1
or
s
i
+
1
s_{i+1}
s
i
+
1
(if
i
>
1
i > 1
i
>
1
and
i
<
40
i < 40
i
<
40
, respectively). He grades each problem exactly once, starting with the first problem of
s
1
s_1
s
1
and ending with the third problem of
s
40
s_{40}
s
40
. Let
N
N
N
be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when
N
N
N
is divided by
100
100
100
.
2012 PUMaC Algebra A7 / B8
Let
a
n
a_n
a
n
be a sequence such that
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
=
⌊
a
n
+
a
n
+
1
2
⌋
a_{n+1} = \lfloor a_n +\sqrt{a_n} +\frac12 \rfloor
a
n
+
1
=
⌊
a
n
+
a
n
+
2
1
⌋
, where
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the greatest integer less than or equal to
x
x
x
. What are the last four digits of
a
2012
a_{2012}
a
2012
?
A5 / B7
1
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2012 PUMaC Combinatorics A5 / B7
5
5
5
people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that
m
n
\frac{m}{n}
n
m
is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where
m
m
m
and
n
n
n
are relatively prime positive integers. What is
m
+
n
m + n
m
+
n
?
A3 / B5
1
Hide problems
2012 PUMaC Combinatorics A3 / B5
Jim has two fair
6
6
6
-sided dice, one whose faces are labelled from
1
1
1
to
6
6
6
, and the second whose faces are labelled from
3
3
3
to
8
8
8
. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is
9
9
9
, if
m
n
\frac{m}{n}
n
m
is the probability he rolled the same die twice where
m
,
n
m, n
m
,
n
are relatively prime positive integers, then what is
m
+
n
m + n
m
+
n
?
A2 / B3
1
Hide problems
2012 PUMaC Combinatorics A2 / B3
How many ways are there to arrange the
6
6
6
permutations of the tuple
(
1
,
2
,
3
)
(1, 2, 3)
(
1
,
2
,
3
)
in a sequence, such that each pair of adjacent permutations contains at least one entry in common? For example, a valid such sequence is given by
(
3
,
2
,
1
)
−
(
2
,
3
,
1
)
−
(
2
,
1
,
3
)
−
(
1
,
2
,
3
)
−
(
1
,
3
,
2
)
−
(
3
,
1
,
2
)
(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)
(
3
,
2
,
1
)
−
(
2
,
3
,
1
)
−
(
2
,
1
,
3
)
−
(
1
,
2
,
3
)
−
(
1
,
3
,
2
)
−
(
3
,
1
,
2
)
.
A1 / B2
1
Hide problems
2012 PUMaC Combinatorics A1 / B2
If the probability that the sum of three distinct integers between
16
16
16
and
30
30
30
(inclusive) is even can be written as
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers, find
m
+
n
m + n
m
+
n
.
A1
2
Hide problems
2012 PUMaC Algebra A1
Compute the smallest positive integer
a
a
a
for which
a
+
a
+
.
.
.
−
1
a
+
1
a
+
.
.
.
>
7
\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7
a
+
a
+
...
−
a
+
a
+
...
1
1
>
7
2012 PUMaC Individual Finals A1
Let
p
p
p
be a prime number greater than
5
5
5
. Prove that there exists a positive integer
n
n
n
such that
p
p
p
divides
2
0
n
+
1
5
n
−
1
2
n
20^n+ 15^n-12^n
2
0
n
+
1
5
n
−
1
2
n
.
A2
1
Hide problems
2012 PUMaC Individual Finals A2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers such that
a
+
b
+
c
=
a
b
c
a+b+c=abc
a
+
b
+
c
=
ab
c
. Prove that
1
a
2
+
1
+
1
b
2
+
1
+
1
c
2
+
1
≥
3
4
\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge \frac{3}{4}
a
2
+
1
1
+
b
2
+
1
1
+
c
2
+
1
1
≥
4
3
.
B1
5
Show problems
A6
4
Show problems
B8
1
Hide problems
2012 PUMaC Geometry B8
A cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
has side lengths
A
B
=
3
,
B
C
=
A
D
=
5
AB = 3, BC = AD = 5
A
B
=
3
,
BC
=
A
D
=
5
, and
C
D
=
8
CD = 8
C
D
=
8
. The radius of its circumcircle can be written in the form
a
b
/
c
a\sqrt{b}/c
a
b
/
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers,
a
,
c
a, c
a
,
c
are relatively prime, and
b
b
b
is not divisible by the square of any prime. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
.
B7
1
Hide problems
2012 PUMaC Geometry B7
Assume the earth is a perfect sphere with a circumference of
60
60
60
units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a
90
90
90
degree angle. If each track has a train of length
L
L
L
traveling at the same speed, what is the maximum value of
L
L
L
such that the trains can travel without crashing?
B3
4
Show problems
B2
4
Show problems
A8
4
Show problems
A7
2
Hide problems
2012 PUMaC Geometry A7
An octahedron (a solid with 8 triangular faces) has a volume of
1040
1040
1040
. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the cyclic quadrilateral. Given that the side lengths of the cyclic quadrilateral are
7
,
15
,
24
,
20
7, 15, 24, 20
7
,
15
,
24
,
20
, in counterclockwise order, the sum of the edge lengths of the entire octahedron can be written in simplest form as
a
/
b
a/b
a
/
b
. Find
a
+
b
a + b
a
+
b
.
2012 PUMaC Number Theory A7
Let
a
,
b
a, b
a
,
b
, and
c
c
c
be positive integers satisfying
a
4
+
a
2
b
2
+
b
4
=
9633
a^4 + a^2b^2 + b^4 = 9633
a
4
+
a
2
b
2
+
b
4
=
9633
2
a
2
+
a
2
b
2
+
2
b
2
+
c
5
=
3605
2a^2 + a^2b^2 + 2b^2 + c^5 = 3605
2
a
2
+
a
2
b
2
+
2
b
2
+
c
5
=
3605
. What is the sum of all distinct values of
a
+
b
+
c
a + b + c
a
+
b
+
c
?
A5
3
Hide problems
2012 PUMaC Geometry A5
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
∠
B
A
C
=
4
5
o
,
∠
B
C
A
=
3
0
o
\angle BAC = 45^o, \angle BCA = 30^o
∠
B
A
C
=
4
5
o
,
∠
BC
A
=
3
0
o
, and
A
B
=
1
AB = 1
A
B
=
1
. Point
D
D
D
lies on segment
A
C
AC
A
C
such that
A
B
=
B
D
AB = BD
A
B
=
B
D
. Find the square of the length of the common external tangent to the circumcircles of triangles
△
B
D
C
\vartriangle BDC
△
B
D
C
and
△
A
B
C
\vartriangle ABC
△
A
BC
.
2012 PUMaC Algebra A5
What is the smallest natural number
n
n
n
greater than
2012
2012
2012
such that the polynomial
f
(
x
)
=
(
x
6
+
x
4
)
n
−
x
4
n
−
x
6
f(x) =(x^6 + x^4)^n - x^{4n} - x^6
f
(
x
)
=
(
x
6
+
x
4
)
n
−
x
4
n
−
x
6
is divisible by
g
(
x
)
=
x
4
+
x
2
+
1
g(x) = x^4 + x^2 + 1
g
(
x
)
=
x
4
+
x
2
+
1
?
2012 PUMaC Number Theory A5
Call a positive integer
x
x
x
a leader if there exists a positive integer
n
n
n
such that the decimal representation of
x
n
x^n
x
n
starts (not ends) with
2012
2012
2012
. For example,
586
586
586
is a leader since
58
6
3
=
201230056
586^3 =201230056
58
6
3
=
201230056
. How many leaders are there in the set
{
1
,
2
,
3
,
.
.
.
,
2012
}
\{1, 2, 3, ..., 2012\}
{
1
,
2
,
3
,
...
,
2012
}
?
A4 / B6
2
Hide problems
2012 PUMaC Geometry A4 / B6
A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of
4
4
4
. The square of the length of the minor axis of the ellipse can be written in the form
a
+
b
c
a + b\sqrt{c}
a
+
b
c
where
a
,
b
a, b
a
,
b
, and
c
c
c
are integers, and
c
c
c
is not divisible by the square of any prime. Find the sum
a
+
b
+
c
a + b + c
a
+
b
+
c
.
2012 PUMaC Combinatorics A4 / B6
How many (possibly empty) sets of lattice points
{
P
1
,
P
2
,
.
.
.
,
P
M
}
\{P_1, P_2, ... , P_M\}
{
P
1
,
P
2
,
...
,
P
M
}
, where each point
P
i
=
(
x
i
,
y
i
)
P_i =(x_i, y_i)
P
i
=
(
x
i
,
y
i
)
for
x
i
,
y
i
∈
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
x_i , y_i \in \{0, 1, 2, 3, 4, 5, 6\}
x
i
,
y
i
∈
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
, satisfy that the slope of the line
P
i
P
j
P_iP_j
P
i
P
j
is positive for each
1
≤
i
<
j
≤
M
1 \le i < j \le M
1
≤
i
<
j
≤
M
? An infinite slope, e.g.
P
i
P_i
P
i
is vertically above
P
j
P_j
P
j
, does not count as positive.
A3
3
Hide problems
2012 PUMaC Geometry A3
Six ants are placed on the vertices of a regular hexagon with an area of
12
12
12
. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance,
s
s
s
, to the next ant. Each ant then proceeds towards the next ant at a speed of
s
100
\frac{s}{100}
100
s
units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of
4
4
4
. T is of the form
a
ln
b
a \ln b
a
ln
b
, where
b
b
b
is square-free. Find
a
+
b
a + b
a
+
b
.
2012 PUMaC Number Theory A3
Let the sequence
{
x
n
}
\{x_n\}
{
x
n
}
be defined by
x
1
∈
{
5
,
7
}
x_1 \in \{5, 7\}
x
1
∈
{
5
,
7
}
and, for
k
≥
1
,
x
k
+
1
∈
{
5
x
k
,
7
x
k
}
k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}
k
≥
1
,
x
k
+
1
∈
{
5
x
k
,
7
x
k
}
. For example, the possible values of
x
3
x_3
x
3
are
5
5
5
,
5
5
7
,
5
7
5
,
5
7
7
,
7
5
5
,
7
5
7
,
7
7
5
5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}
5
5
5
,
5
5
7
,
5
7
5
,
5
7
7
,
7
5
5
,
7
5
7
,
7
7
5
, and
7
7
7
7^{7^7}
7
7
7
. Determine the sum of all possible values for the last two digits of
x
2012
x_{2012}
x
2012
.
2012 PUMaC Individual Finals A3
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
, and let
D
D
D
be the foot of the angle bisector from
A
A
A
to
B
C
BC
BC
. Let
Γ
\Gamma
Γ
be the circumcircle of triangle
B
I
C
BIC
B
I
C
, and let
P
Q
PQ
PQ
be a chord of
Γ
\Gamma
Γ
passing through
D
D
D
. Prove that
A
D
AD
A
D
bisects
∠
P
A
Q
\angle PAQ
∠
P
A
Q
.
A2 / B5
2
Hide problems
2012 PUMaC Geometry A2 / B5
Two circles centered at
O
O
O
and
P
P
P
have radii of length
5
5
5
and
6
6
6
respectively. Circle
O
O
O
passes through point
P
P
P
. Let the intersection points of circles
O
O
O
and
P
P
P
be
M
M
M
and
N
N
N
. The area of triangle
△
M
N
P
\vartriangle MNP
△
MNP
can be written in simplest form as
a
/
b
a/b
a
/
b
. Find
a
+
b
a + b
a
+
b
.
2012 PUMaC Number Theory A2 / B5
How many ways can
2
2012
2^{2012}
2
2012
be expressed as the sum of four (not necessarily distinct) positive squares?
A1 / B4
2
Hide problems
2012 PUMaC Geometry A1 / B4
Three circles, with radii of
1
,
1
1, 1
1
,
1
, and
2
2
2
, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as
a
+
b
c
a + b\sqrt{c}
a
+
b
c
where
c
c
c
is not divisible by any perfect square larger than
1
1
1
. Find
a
+
b
+
c
a + b + c
a
+
b
+
c
2012 PUMaC Number Theory A1 / B4
Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his
20
20
20
friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over
500
500
500
candies, what is the fewest number of candies the bag can contain?