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Problems
Contests
Undergraduate contests
Putnam
2006 Putnam
2006 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 2006 B6
Let
k
k
k
be an integer greater than
1.
1.
1.
Suppose
a
0
>
0
a_{0}>0
a
0
>
0
and define
a
n
+
1
=
a
n
+
1
a
n
k
a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}
a
n
+
1
=
a
n
+
k
a
n
1
for
n
≥
0.
n\ge 0.
n
≥
0.
Evaluate
lim
n
→
∞
a
n
k
+
1
n
k
.
\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.
n
→
∞
lim
n
k
a
n
k
+
1
.
B5
1
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Putnam 2006 B5
For each continuous function
f
:
[
0
,
1
]
→
R
,
f: [0,1]\to\mathbb{R},
f
:
[
0
,
1
]
→
R
,
let
I
(
f
)
=
∫
0
1
x
2
f
(
x
)
d
x
I(f)=\int_{0}^{1}x^{2}f(x)\,dx
I
(
f
)
=
∫
0
1
x
2
f
(
x
)
d
x
and
J
(
f
)
=
∫
0
1
x
(
f
(
x
)
)
2
d
x
.
J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.
J
(
f
)
=
∫
0
1
x
(
f
(
x
)
)
2
d
x
.
Find the maximum value of
I
(
f
)
−
J
(
f
)
I(f)-J(f)
I
(
f
)
−
J
(
f
)
over all such functions
f
.
f.
f
.
B4
1
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Putnam 2006 B4
Let
Z
Z
Z
denote the set of points in
R
n
\mathbb{R}^{n}
R
n
whose coordinates are
0
0
0
or
1.
1.
1.
(Thus
Z
Z
Z
has
2
n
2^{n}
2
n
elements, which are the vertices of a unit hypercube in
R
n
\mathbb{R}^{n}
R
n
.) Given a vector subspace
V
V
V
of
R
n
,
\mathbb{R}^{n},
R
n
,
let
Z
(
V
)
Z(V)
Z
(
V
)
denote the number of members of
Z
Z
Z
that lie in
V
.
V.
V
.
Let
k
k
k
be given,
0
≤
k
≤
n
.
0\le k\le n.
0
≤
k
≤
n
.
Find the maximum, over all vector subspaces
V
⊆
R
n
V\subseteq\mathbb{R}^{n}
V
⊆
R
n
of dimension
k
,
k,
k
,
of the number of points in
V
∩
Z
.
V\cap Z.
V
∩
Z
.
B3
1
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Putnam 2006 B3
Let
S
S
S
be a finite set of points in the plane. A linear partition of
S
S
S
is an unordered pair
{
A
,
B
}
\{A,B\}
{
A
,
B
}
of subsets of
S
S
S
such that
A
∪
B
=
S
,
A
∩
B
=
∅
,
A\cup B=S,\ A\cap B=\emptyset,
A
∪
B
=
S
,
A
∩
B
=
∅
,
and
A
A
A
and
B
B
B
lie on opposite sides of some straight line disjoint from
S
S
S
(
A
A
A
or
B
B
B
may be empty). Let
L
S
L_{S}
L
S
be the number of linear partitions of
S
.
S.
S
.
For each positive integer
n
,
n,
n
,
find the maximum of
L
S
L_{S}
L
S
over all sets
S
S
S
of
n
n
n
points.
B2
1
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Putnam 2006 B2
Prove that, for every set
X
=
{
x
1
,
x
2
,
…
,
x
n
}
X=\{x_{1},x_{2},\dots,x_{n}\}
X
=
{
x
1
,
x
2
,
…
,
x
n
}
of
n
n
n
real numbers, there exists a non-empty subset
S
S
S
of
X
X
X
and an integer
m
m
m
such that
∣
m
+
∑
s
∈
S
s
∣
≤
1
n
+
1
\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}
m
+
s
∈
S
∑
s
≤
n
+
1
1
B1
1
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Putnam 2006 B1
Show that the curve
x
3
+
3
x
y
+
y
3
=
1
x^{3}+3xy+y^{3}=1
x
3
+
3
x
y
+
y
3
=
1
contains only one set of three distinct points,
A
,
B
,
A,B,
A
,
B
,
and
C
,
C,
C
,
which are the vertices of an equilateral triangle.
A6
1
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Putnam 2006 A6
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
A5
1
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Putnam 2006 A5
Let
n
n
n
be a positive odd integer and let
θ
\theta
θ
be a real number such that
θ
/
π
\theta/\pi
θ
/
π
is irrational. Set
a
k
=
tan
(
θ
+
k
π
/
n
)
,
k
=
1
,
2
…
,
n
.
a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.
a
k
=
tan
(
θ
+
kπ
/
n
)
,
k
=
1
,
2
…
,
n
.
Prove that
a
1
+
a
2
+
⋯
+
a
n
a
1
a
2
⋯
a
n
\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}
a
1
a
2
⋯
a
n
a
1
+
a
2
+
⋯
+
a
n
is an integer, and determine its value.
A4
1
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Putnam 2006 A4
Let
S
=
{
1
,
2
…
,
n
}
S=\{1,2\dots,n\}
S
=
{
1
,
2
…
,
n
}
for some integer
n
>
1.
n>1.
n
>
1.
Say a permutation
π
\pi
π
of
S
S
S
has a local maximum at
k
∈
S
k\in S
k
∈
S
if
(i)
π
(
k
)
>
π
(
k
+
1
)
for
k
=
1
(ii)
π
(
k
−
1
)
<
π
(
k
)
and
π
(
k
)
>
π
(
k
+
1
)
for
1
<
k
<
n
(iii)
π
(
k
−
1
)
M
π
(
k
)
for
k
=
n
\begin{array}{ccc}\text{(i)}&\pi(k)>\pi(k+1)&\text{for }k=1\\ \text{(ii)}&\pi(k-1)<\pi(k)\text{ and }\pi(k)>\pi(k+1)&\text{for }1<k<n\\ \text{(iii)}&\pi(k-1)M\pi(k)&\text{for }k=n\end{array}
(i)
(ii)
(iii)
π
(
k
)
>
π
(
k
+
1
)
π
(
k
−
1
)
<
π
(
k
)
and
π
(
k
)
>
π
(
k
+
1
)
π
(
k
−
1
)
M
π
(
k
)
for
k
=
1
for
1
<
k
<
n
for
k
=
n
(For example, if
n
=
5
n=5
n
=
5
and
π
\pi
π
takes values at
1
,
2
,
3
,
4
,
5
1,2,3,4,5
1
,
2
,
3
,
4
,
5
of
2
,
1
,
4
,
5
,
3
,
2,1,4,5,3,
2
,
1
,
4
,
5
,
3
,
then
π
\pi
π
has a local maximum of
2
2
2
as
k
=
1
,
k=1,
k
=
1
,
and a local maximum at
k
−
4.
k-4.
k
−
4.
) What is the average number of local maxima of a permutation of
S
,
S,
S
,
averaging over all permuatations of
S
?
S?
S
?
A3
1
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Putnam 2006 A3
Let
1
,
2
,
3
,
…
,
2005
,
2006
,
2007
,
2009
,
2012
,
2016
,
…
1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots
1
,
2
,
3
,
…
,
2005
,
2006
,
2007
,
2009
,
2012
,
2016
,
…
be a sequence defined by
x
k
=
k
x_{k}=k
x
k
=
k
for
k
=
1
,
2
…
,
2006
k=1,2\dots,2006
k
=
1
,
2
…
,
2006
and
x
k
+
1
=
x
k
+
x
k
−
2005
x_{k+1}=x_{k}+x_{k-2005}
x
k
+
1
=
x
k
+
x
k
−
2005
for
k
≥
2006.
k\ge 2006.
k
≥
2006.
Show that the sequence has 2005 consecutive terms each divisible by 2006.
A2
1
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Putnam 2006 A2
Alice and Bob play a game in which they take turns removing stones from a heap that initially has
n
n
n
stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such
n
n
n
such that Bob has a winning strategy. (For example, if
n
=
17
,
n=17,
n
=
17
,
then Alice might take
6
6
6
leaving
11
;
11;
11
;
then Bob might take
1
1
1
leaving
10
;
10;
10
;
then Alice can take the remaining stones to win.)
A1
1
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Putnam 2006 A1
Find the volume of the region of points
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
(
x
2
+
y
2
+
z
2
+
8
)
2
≤
36
(
x
2
+
y
2
)
.
\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right).
(
x
2
+
y
2
+
z
2
+
8
)
2
≤
36
(
x
2
+
y
2
)
.