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Putnam
1955 Putnam
1955 Putnam
Part of
Putnam
Subcontests
(14)
B7
1
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Putnam 1955 B7
Four forces acting on a body are in equilibrium. Prove that, if their lines of action are mutually skew, they are rulings of a hyperboloid.
B6
1
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Putnam 1955 B6
Prove: If
f
(
x
)
>
0
f(x) > 0
f
(
x
)
>
0
for all
x
x
x
and
f
(
x
)
→
0
f(x) \rightarrow 0
f
(
x
)
→
0
as
x
→
∞
,
x \rightarrow \infty,
x
→
∞
,
then there exists at most a finite number of solutions of
f
(
m
)
+
f
(
n
)
+
f
(
p
)
=
1
f(m) + f(n) + f(p) = 1
f
(
m
)
+
f
(
n
)
+
f
(
p
)
=
1
in positive integers
m
,
n
,
m, n,
m
,
n
,
and
p
.
p.
p
.
B5
1
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Putnam 1955 B5
Given an infinite sequence of
0
0
0
's and
1
1
1
's and a fixed integer
k
,
k,
k
,
suppose that there are no more than
k
k
k
distinct blocks of
k
k
k
consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence
11011010101
11011010101
11011010101
followed by alternating
0
0
0
's and
1
1
1
's indefinitely, which is periodic beginning with the fifth term.)
B4
1
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Putnam 1955 B4
Do there exist
1
,
000
,
000
1,000,000
1
,
000
,
000
consecutive integers each of which contains a repeated prime factor?
B3
1
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Putnam 1955 B3
Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)
B2
1
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Putnam 1955 B2
Suppose that
f
f
f
is a function with two continuous derivatives 2and
f
(
0
)
=
0.
f(0) = 0.
f
(
0
)
=
0.
Prove that the function
g
,
g,
g
,
defined by
g
(
0
)
=
f
′
(
0
)
,
g
(
x
)
=
f
(
x
)
/
x
g(0) = f '(0), g(x) = f(x) / x
g
(
0
)
=
f
′
(
0
)
,
g
(
x
)
=
f
(
x
)
/
x
for
x
≠
0
,
x \ne 0,
x
=
0
,
has a continuous derivative.
B1
1
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Putnam 1955 B1
A sphere rolls along two intersecting straight lines. Find the locus of its center.
A7
1
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Putnam 1955 A7
Consider the function
f
f
f
defined by the differential equation
f
′
′
(
x
)
=
(
x
3
+
a
x
)
f
(
x
)
f'' (x) = (x^3 + ax) f(x)
f
′′
(
x
)
=
(
x
3
+
a
x
)
f
(
x
)
and the initial conditions
f
(
0
)
=
1
,
f
′
(
0
)
=
0.
f(0) = 1, f'(0) = 0.
f
(
0
)
=
1
,
f
′
(
0
)
=
0.
Prove that the roots of
f
f
f
are bounded above but unbounded below.
A6
1
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Putnam 1955 A6
Find a necessary and sufficient condition on the positive integer
n
n
n
that the equation
x
n
+
(
2
+
x
)
n
+
(
2
−
x
)
n
=
0
x^n + (2 + x)^n + (2 - x)^n = 0
x
n
+
(
2
+
x
)
n
+
(
2
−
x
)
n
=
0
have a rational root.
A5
1
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Putnam 1955 A5
If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.
A4
1
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Putnam 1955 A4
On a circle,
n
n
n
points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?
A3
1
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Putnam 1955 A3
Suppose that
∑
i
=
1
∞
x
i
\sum^\infty_{i=1} x_i
∑
i
=
1
∞
x
i
is a convergent series of positive terms which monotonically decrease (that is,
x
1
≥
x
2
≥
x
3
≥
⋯
x_1 \ge x_2 \ge x_3 \ge \cdots
x
1
≥
x
2
≥
x
3
≥
⋯
). Let
P
P
P
denote the set of all numbers which are sums of some (finite or infinite) subseries of
∑
i
=
1
∞
x
i
.
\sum^\infty_{i= 1} x_i.
∑
i
=
1
∞
x
i
.
Show that
P
P
P
is an interval if and only if
x
n
≤
∑
i
=
n
+
1
∞
x
i
x_n \le \sum^\infty_{i = n + 1} x_i
x
n
≤
i
=
n
+
1
∑
∞
x
i
for every integer
n
.
n.
n
.
A2
1
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Putnam 1955 A2
A
1
A
2
…
A
n
A_1 ~A_2~ \ldots ~A_n
A
1
A
2
…
A
n
is a regular polygon inscribed in a circle of radius
r
r
r
and center
O
.
O.
O
.
P
P
P
is a point on line
O
A
1
OA_1
O
A
1
extended beyond
A
1
.
A_1.
A
1
.
Show that
∏
i
=
1
n
P
A
‾
i
=
O
P
‾
n
−
r
n
.
\prod^n_{i=1} ~ \overline{PA}_{~i} = \overline{OP}^{~n} - r^n.
i
=
1
∏
n
P
A
i
=
OP
n
−
r
n
.
A1
1
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Putnam 1955 A1
Prove that there is no set of integers
m
,
n
,
p
m, n, p
m
,
n
,
p
except
0
,
0
,
0
0, 0, 0
0
,
0
,
0
for which
m
+
n
2
+
p
3
=
0.
m + n \sqrt2 + p \sqrt3 = 0.
m
+
n
2
+
p
3
=
0.