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Putnam
1952 Putnam
1952 Putnam
Part of
Putnam
Subcontests
(14)
B7
1
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Putnam 1952 B7
Given any real number
N
0
,
N_0,
N
0
,
if
N
j
+
1
=
cos
N
j
,
N_{j+1}= \cos N_j ,
N
j
+
1
=
cos
N
j
,
prove that
lim
j
→
∞
N
j
\lim_{j\to \infty} N_j
lim
j
→
∞
N
j
exists and is independent of
N
0
.
N_0.
N
0
.
B6
1
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Putnam 1952 B6
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
B5
1
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Putnam 1952 B5
If the terms of a sequence
a
1
,
a
2
,
…
a_{1}, a_{2}, \ldots
a
1
,
a
2
,
…
are monotonic, and if
∑
n
=
1
∞
a
n
\sum_{n=1}^{\infty} a_n
∑
n
=
1
∞
a
n
converges, show that
∑
n
=
1
∞
n
(
a
n
−
a
n
+
1
)
\sum_{n=1}^{\infty} n(a_{n} -a_{n+1 })
∑
n
=
1
∞
n
(
a
n
−
a
n
+
1
)
converges.
B4
1
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Putnam 1952 B4
A homogeneous solid body is made by joining a base of a circular cylinder of height
h
h
h
and radius
r
,
r,
r
,
and the base of a hemisphere of radius
r
.
r.
r
.
This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if
r
r
r
is large as compared to
h
h
h
, the equilibrium will be stable; but if
r
r
r
is small compared to
h
h
h
, the equilibrium will be unstable. What is the critical value of the ratio r\slash h which enables the body to rest in neutral equilibrium in any position?
B3
1
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Putnam 1952 B3
Develop necessary and sufficient conditions that the equation
∣
0
a
1
−
x
a
2
−
x
−
a
1
−
x
0
a
3
−
x
−
a
2
−
x
−
a
3
−
x
0
∣
=
0
(
a
i
≠
0
)
\begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0)
0
−
a
1
−
x
−
a
2
−
x
a
1
−
x
0
−
a
3
−
x
a
2
−
x
a
3
−
x
0
=
0
(
a
i
=
0
)
shall have a multiple root.
B2
1
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Putnam 1952 B2
Find the surface generated by the solutions of
d
x
y
z
=
d
y
z
x
=
d
z
x
y
,
\frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy},
yz
d
x
=
z
x
d
y
=
x
y
d
z
,
which intersects the circle
y
2
+
z
2
=
1
,
x
=
0.
y^2+ z^2 = 1, x = 0.
y
2
+
z
2
=
1
,
x
=
0.
B1
1
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Putnam 1952 B1
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines:
a
2
=
b
2
+
c
2
−
2
b
c
cos
A
,
a^2 = b^2 + c^2 - 2bc \cos A,
a
2
=
b
2
+
c
2
−
2
b
c
cos
A
,
to find the third side
a
.
a.
a
.
He uses logarithms as follows. He finds
log
b
\log b
lo
g
b
and doubles it; adds to that the double of
log
c
;
\log c;
lo
g
c
;
subtracts the sum of the logarithms of
2
,
b
,
c
,
2, b, c,
2
,
b
,
c
,
and
cos
A
;
\cos A;
cos
A
;
divides the result by
2
;
2;
2
;
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
A7
1
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Putnam 1952 A7
Directed lines are drawn from the center of a circle, making angles of
0
,
±
1
,
±
2
,
±
3
,
…
0, \pm 1, \pm 2, \pm 3, \ldots
0
,
±
1
,
±
2
,
±
3
,
…
(measured in radians from a prime direction). If these lines meet the circle in points
P
0
,
P
1
,
P
−
1
,
P
2
,
P
−
2
,
…
,
P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots,
P
0
,
P
1
,
P
−
1
,
P
2
,
P
−
2
,
…
,
show that there is no interval on the circumference of the circle which does not contain some
P
±
i
.
P_{\pm i}.
P
±
i
.
(You may assume that
π
\pi
π
is irrational.)
A6
1
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Putnam 1952 A6
A man has a rectangular block of wood
m
m
m
by
n
n
n
by
r
r
r
inches (
m
,
n
,
m, n,
m
,
n
,
and
r
r
r
are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.)
A5
1
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Putnam 1952 A5
Let
a
j
(
j
=
1
,
2
,
…
,
n
)
a_j (j = 1, 2, \ldots, n)
a
j
(
j
=
1
,
2
,
…
,
n
)
be entirely arbitrary numbers except that no one is equal to unity. Prove
a
1
+
∑
i
=
2
n
a
i
∏
j
=
1
i
−
1
(
1
−
a
j
)
=
1
−
∏
j
=
1
n
(
1
−
a
j
)
.
a_1 + \sum^n_{i=2} a_i \prod^{i-1}_{j=1} (1 - a_j) = 1 - \prod^n_{j=1} (1 - a_j).
a
1
+
i
=
2
∑
n
a
i
j
=
1
∏
i
−
1
(
1
−
a
j
)
=
1
−
j
=
1
∏
n
(
1
−
a
j
)
.
A4
1
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Putnam 1952 A4
The flag of the United Nations consists of a polar map of the world, with the North Pole as its center, extending to approximately
4
5
∘
45^\circ
4
5
∘
South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude
3
0
∘
30^\circ
3
0
∘
S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
A3
1
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Putnam 1952 A3
Develop necessary and sufficient conditions which ensure that
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
and
r
1
2
,
r
2
2
,
r
3
2
r_1^2, r_2^2, r_3^2
r
1
2
,
r
2
2
,
r
3
2
are simultaneously roots of the 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.
A2
1
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Putnam 1952 A2
Show that the equation
(
9
−
x
2
)
(
d
y
d
x
)
2
=
(
9
−
y
2
)
(9 - x^2) \left (\frac{\mathrm dy}{\mathrm dx} \right)^2 = (9 - y^2)
(
9
−
x
2
)
(
d
x
d
y
)
2
=
(
9
−
y
2
)
characterizes a family of conics touching the four sides of a fixed square.
A1
1
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Putnam 1952 A1
Let
f
(
x
)
=
∑
i
=
0
i
=
n
a
i
x
n
−
i
f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}
f
(
x
)
=
i
=
0
∑
i
=
n
a
i
x
n
−
i
be a polynomial of degree
n
n
n
with integral coefficients. If
a
0
,
a
n
,
a_0, a_n,
a
0
,
a
n
,
and
f
(
1
)
f(1)
f
(
1
)
are odd, prove that
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
has no rational roots.