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Putnam
1998 Putnam
1998 Putnam
Part of
Putnam
Subcontests
(6)
6
2
Hide problems
1998 Putnam A6
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
denote distinct points with integer coefficients in
R
2
\mathbb{R}^2
R
2
. Prove that if
(
∣
A
B
∣
+
∣
B
C
∣
)
2
<
8
⋅
[
A
B
C
]
+
1
(|AB|+|BC|)^2<8\cdot[ABC]+1
(
∣
A
B
∣
+
∣
BC
∣
)
2
<
8
⋅
[
A
BC
]
+
1
then
A
,
B
,
C
A,B,C
A
,
B
,
C
are three vertices of a square. Here
∣
X
Y
∣
|XY|
∣
X
Y
∣
is the length of segment
X
Y
XY
X
Y
and
[
A
B
C
]
[ABC]
[
A
BC
]
is the area of triangle
A
B
C
ABC
A
BC
.
1998 Putnam B6
Prove that, for any integers
a
,
b
,
c
a,b,c
a
,
b
,
c
, there exists a positive integer
n
n
n
such that
n
3
+
a
n
2
+
b
n
+
c
\sqrt{n^3+an^2+bn+c}
n
3
+
a
n
2
+
bn
+
c
is not an integer.
4
2
Hide problems
1998 Putnam A4
Let
A
1
=
0
A_1=0
A
1
=
0
and
A
2
=
1
A_2=1
A
2
=
1
. For
n
>
2
n>2
n
>
2
, the number
A
n
A_n
A
n
is defined by concatenating the decimal expansions of
A
n
−
1
A_{n-1}
A
n
−
1
and
A
n
−
2
A_{n-2}
A
n
−
2
from left to right. For example
A
3
=
A
2
A
1
=
10
A_3=A_2A_1=10
A
3
=
A
2
A
1
=
10
,
A
4
=
A
3
A
2
=
101
A_4=A_3A_2=101
A
4
=
A
3
A
2
=
101
,
A
5
=
A
4
A
3
=
10110
A_5=A_4A_3=10110
A
5
=
A
4
A
3
=
10110
, and so forth. Determine all
n
n
n
such that
11
11
11
divides
A
n
A_n
A
n
.
1998 Putnam B4
Find necessary and sufficient conditions on positive integers
m
m
m
and
n
n
n
so that
∑
i
=
0
m
n
−
1
(
−
1
)
⌊
i
/
m
⌋
+
⌊
i
/
n
⌋
=
0.
\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.
i
=
0
∑
mn
−
1
(
−
1
)
⌊
i
/
m
⌋
+
⌊
i
/
n
⌋
=
0.
3
2
Hide problems
1998 Putnam A3
Let
f
f
f
be a real function on the real line with continuous third derivative. Prove that there exists a point
a
a
a
such that
f
(
a
)
⋅
f
′
(
a
)
⋅
f
′
′
(
a
)
⋅
f
′
′
′
(
a
)
≥
0.
f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.
f
(
a
)
⋅
f
′
(
a
)
⋅
f
′′
(
a
)
⋅
f
′′′
(
a
)
≥
0.
Putnam 1998 B3
Let
H
H
H
be the unit hemisphere
{
(
x
,
y
,
z
)
:
x
2
+
y
2
+
z
2
=
1
,
z
≥
0
}
\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}
{(
x
,
y
,
z
)
:
x
2
+
y
2
+
z
2
=
1
,
z
≥
0
}
,
C
C
C
the unit circle
{
(
x
,
y
,
0
)
:
x
2
+
y
2
=
1
}
\{(x,y,0):x^2+y^2=1\}
{(
x
,
y
,
0
)
:
x
2
+
y
2
=
1
}
, and
P
P
P
the regular pentagon inscribed in
C
C
C
. Determine the surface area of that portion of
H
H
H
lying over the planar region inside
P
P
P
, and write your answer in the form
A
sin
α
+
B
cos
β
A \sin\alpha + B \cos\beta
A
sin
α
+
B
cos
β
, where
A
,
B
,
α
,
β
A,B,\alpha,\beta
A
,
B
,
α
,
β
are real numbers.
2
2
Hide problems
Putnam 1998 A2
Let
s
s
s
be any arc of the unit circle lying entirely in the first quadrant. Let
A
A
A
be the area of the region lying below
s
s
s
and above the
x
x
x
-axis and let
B
B
B
be the area of the region lying to the right of the
y
y
y
-axis and to the left of
s
s
s
. Prove that
A
+
B
A+B
A
+
B
depends only on the arc length, and not on the position, of
s
s
s
.
1998 Putnam B2
Given a point
(
a
,
b
)
(a,b)
(
a
,
b
)
with
0
<
b
<
a
0<b<a
0
<
b
<
a
, determine the minimum perimeter of a triangle with one vertex at
(
a
,
b
)
(a,b)
(
a
,
b
)
, one on the
x
x
x
-axis, and one on the line
y
=
x
y=x
y
=
x
. You may assume that a triangle of minimum perimeter exists.
1
2
Hide problems
Putnam 1998 A1
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?
1998 Putnam B1
Find the minimum value of
(
x
+
1
/
x
)
6
−
(
x
6
+
1
/
x
6
)
−
2
(
x
+
1
/
x
)
3
+
(
x
3
+
1
/
x
3
)
\dfrac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}
(
x
+
1/
x
)
3
+
(
x
3
+
1/
x
3
)
(
x
+
1/
x
)
6
−
(
x
6
+
1/
x
6
)
−
2
for
x
>
0
x>0
x
>
0
.
5
2
Hide problems
1998 Putnam A5
Let
F
\mathcal{F}
F
be a finite collection of open discs in
R
2
\mathbb{R}^2
R
2
whose union contains a set
E
⊆
R
2
E\subseteq \mathbb{R}^2
E
⊆
R
2
. Show that there is a pairwise disjoint subcollection
D
1
,
…
,
D
n
D_1,\ldots,D_n
D
1
,
…
,
D
n
in
F
\mathcal{F}
F
such that
E
⊆
∪
j
=
1
n
3
D
j
.
E\subseteq\cup_{j=1}^n 3D_j.
E
⊆
∪
j
=
1
n
3
D
j
.
Here, if
D
D
D
is the disc of radius
r
r
r
and center
P
P
P
, then
3
D
3D
3
D
is the disc of radius
3
r
3r
3
r
and center
P
P
P
.
Putnam 1998 B5
Let
N
N
N
be the positive integer with 1998 decimal digits, all of them 1; that is,
N
=
1111
⋯
11.
N=1111\cdots 11.
N
=
1111
⋯
11.
Find the thousandth digit after the decimal point of
N
\sqrt N
N
.