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Putnam
1951 Putnam
1951 Putnam
Part of
Putnam
Subcontests
(14)
B7
1
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Putnam 1951 B7
Find the volume of the four-dimensional hypersphere
x
2
+
y
2
+
z
2
+
t
2
=
r
2
x^2 +y^2 +z^2 +t^2 =r^2
x
2
+
y
2
+
z
2
+
t
2
=
r
2
and the hypervolume of its interior
x
2
+
y
2
+
z
2
+
t
2
<
r
2
x^2 +y^2 +z^2 +t^2 <r^2
x
2
+
y
2
+
z
2
+
t
2
<
r
2
B6
1
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Putnam 1951 B6
Assuming that all of the roots of the cubic equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 +bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
are real, show that the difference between the greatest and the least roots is not less than
(
a
2
−
3
b
)
1
/
2
(a^2 - 3b)^{1/2}
(
a
2
−
3
b
)
1/2
or greater than
2
(
a
2
−
3
b
)
1
/
2
/
3
1
/
2
.
2 (a^2 - 3b)^{1/2} / 3^{1/2}.
2
(
a
2
−
3
b
)
1/2
/
3
1/2
.
B5
1
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Putnam 1951 B5
A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.
B4
1
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Putnam 1951 B4
Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.
B3
1
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Putnam 1951 B3
Show that if
x
x
x
is positive, then
log
e
(
1
+
1
/
x
)
>
1
/
(
1
+
x
)
.
\log_e (1 + 1/x) > 1 / (1 + x).
lo
g
e
(
1
+
1/
x
)
>
1/
(
1
+
x
)
.
B2
1
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Putnam 1951 B2
Two functions of
x
x
x
are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.
B1
1
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Putnam 1951 B1
Find the conditions that the functions
M
(
x
,
y
)
M(x, y)
M
(
x
,
y
)
and
N
(
x
,
y
)
N (x, y)
N
(
x
,
y
)
must satisfy in order that the differential equation
M
d
x
+
N
d
y
=
0
Mdx + Ndy =0
M
d
x
+
N
d
y
=
0
shall have an integrating factor of the form
f
(
x
y
)
.
f(xy).
f
(
x
y
)
.
You may assume that
M
M
M
and
N
N
N
have continuous partial derivatives of all orders.
A7
1
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Putnam 1951 A7
Show that if the series
a
1
+
a
2
+
a
3
+
⋯
+
a
n
+
⋯
a_1 + a_2 + a_3 + \cdots + a_n + \cdots
a
1
+
a
2
+
a
3
+
⋯
+
a
n
+
⋯
converges, then the series
a
1
+
a
2
/
2
+
a
3
/
3
+
⋯
+
a
n
/
n
+
⋯
a_1 + a_2 / 2 + a_3 / 3 + \cdots + a_n / n + \cdots
a
1
+
a
2
/2
+
a
3
/3
+
⋯
+
a
n
/
n
+
⋯
converges also.
A6
1
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Putnam 1951 A6
Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.
A5
1
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Putnam 1951 A5
Consider in the plane the network of points having integral coordinates. For lines having rational slope show that:(i) the line passes through no points of the network or through infinitely many;(ii) there exists for each line a positive number
d
d
d
having the property that no point of the network, except such as may be on the line, is closer to the line than the distance
d
.
d.
d
.
A4
1
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Putnam 1951 A4
Trace the curve whose equation is:
y
4
−
x
4
−
96
y
2
+
100
x
2
=
0.
y^4 - x^4 - 96y^2 + 100x^2 = 0.
y
4
−
x
4
−
96
y
2
+
100
x
2
=
0.
A3
1
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Putnam 1951 A3
Find the sum to infinity of the series:
1
−
1
4
+
1
7
−
1
10
+
⋯
+
(
−
1
)
n
+
1
3
n
−
2
+
⋯
.
1 - \frac 14 + \frac 17 - \frac 1{10} + \cdots + \frac{(-1)^{n + 1}}{3n - 2} + \cdots.
1
−
4
1
+
7
1
−
10
1
+
⋯
+
3
n
−
2
(
−
1
)
n
+
1
+
⋯
.
A2
1
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Putnam 1951 A2
In the plane, what is the locus of points of the sum of the squares of whose distances from
n
n
n
fixed points is a constant? What restrictions, stated in geometric terms, must be put on the constant so that the locus is non-null?
A1
1
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Putnam 1951 A1
Show that the determinant:
∣
0
a
b
c
−
a
0
d
e
−
b
−
d
0
f
−
c
−
e
−
f
0
∣
\begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{vmatrix}
0
−
a
−
b
−
c
a
0
−
d
−
e
b
d
0
−
f
c
e
f
0
is non-negative, if its elements
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
etc., are real.