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Putnam
1956 Putnam
1956 Putnam
Part of
Putnam
Subcontests
(14)
B7
1
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Putnam 1956 B7
The polynomials
P
(
z
)
P(z)
P
(
z
)
and
Q
(
z
)
Q(z)
Q
(
z
)
with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
P
(
z
)
+
1
and
Q
(
z
)
+
1.
P(z)+1 \;\; \text{and} \;\; Q(z)+1.
P
(
z
)
+
1
and
Q
(
z
)
+
1.
Prove that
P
(
z
)
=
Q
(
z
)
.
P(z)=Q(z).
P
(
z
)
=
Q
(
z
)
.
B6
1
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Putnam 1956 B6
Given
T
1
=
2
,
T
n
+
1
=
T
n
2
−
T
n
+
1
T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1
T
1
=
2
,
T
n
+
1
=
T
n
2
−
T
n
+
1
for
n
>
0.
n>0.
n
>
0.
Prove:(i) If
m
≠
n
,
m \ne n,
m
=
n
,
T
m
T_m
T
m
and
T
n
T_n
T
n
have no common factor greater than
1.
1.
1.
(ii)
∑
i
=
1
∞
1
T
i
=
1.
\sum_{i=1}^{\infty} \frac{1}{T_i }=1.
∑
i
=
1
∞
T
i
1
=
1.
B4
1
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Putnam 1956 B4
Prove that if
A
,
B
,
A,B,
A
,
B
,
and
C
C
C
are angles of a triangle measured in radians then
A
cos
B
+
sin
A
cos
C
>
0.
A \cos B +\sin A \cos C >0.
A
cos
B
+
sin
A
cos
C
>
0.
B3
1
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Putnam 1956 B3
A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.
B2
1
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Putnam 1956 B2
Suppose that each set
X
X
X
of points in the plane has an associated set
X
‾
\overline{X}
X
of points called its cover. Suppose further that (1)
X
∪
Y
‾
⊃
X
‾
‾
∪
Y
‾
∪
Y
\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y
X
∪
Y
⊃
X
∪
Y
∪
Y
for all sets
X
,
Y
X,Y
X
,
Y
. Show that i)
X
‾
⊃
X
\overline{X} \supset X
X
⊃
X
, ii)
X
‾
‾
=
X
‾
\overline{\overline{X}}=\overline{X}
X
=
X
and iii)
X
⊃
Y
⇒
X
‾
⊃
Y
‾
.
X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.
X
⊃
Y
⇒
X
⊃
Y
.
Prove also that these three statements imply (1).
B1
1
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Putnam 1956 B1
Show that if the differential equation
M
(
x
,
y
)
d
x
+
N
(
x
,
y
)
d
y
=
0
M(x,y)\, dx +N(x,y) \, dy =0
M
(
x
,
y
)
d
x
+
N
(
x
,
y
)
d
y
=
0
is both homogeneous and exact, then the solution
y
=
y
(
x
)
y=y(x)
y
=
y
(
x
)
satisfies that
x
M
(
x
,
y
)
+
y
N
(
x
,
y
)
xM(x,y)+yN(x,y)
x
M
(
x
,
y
)
+
y
N
(
x
,
y
)
is constant.
A7
1
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Putnam 1956 A7
Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of
2.
2.
2.
A6
1
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Putnam 1956 A6
i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances. ii) Show that the corresponding statement for the line is false.
A5
1
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Putnam 1956 A5
Call a subset of
{
1
,
2
,
…
,
n
}
\{1,2,\ldots, n\}
{
1
,
2
,
…
,
n
}
unfriendly if no two of its elements are consecutive. Show that the number of unfriendly subsets with
k
k
k
elements is
(
n
−
k
+
1
k
)
.
\binom{n-k+1}{k}.
(
k
n
−
k
+
1
)
.
A4
1
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Putnam 1956 A4
Suppose that the
n
n
n
times differentiable real function
f
(
x
)
f(x)
f
(
x
)
has at least
n
+
1
n+1
n
+
1
distinct zeros in the closed interval
[
a
,
b
]
[a,b]
[
a
,
b
]
and that the polynomial
P
(
z
)
=
z
n
+
c
n
−
1
z
n
−
1
+
…
+
c
1
x
+
c
0
P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0
P
(
z
)
=
z
n
+
c
n
−
1
z
n
−
1
+
…
+
c
1
x
+
c
0
has only real zeroes. Show that
f
(
n
)
(
x
)
+
c
n
−
1
f
(
n
−
1
)
(
x
)
+
…
+
c
1
f
′
(
x
)
+
c
0
f
(
x
)
f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)
f
(
n
)
(
x
)
+
c
n
−
1
f
(
n
−
1
)
(
x
)
+
…
+
c
1
f
′
(
x
)
+
c
0
f
(
x
)
has at least one zero in
[
a
,
b
]
[a,b]
[
a
,
b
]
, where
f
(
n
)
f^{(n)}
f
(
n
)
denotes the
n
n
n
-th derivative of
f
.
f.
f
.
A3
1
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Putnam 1956 A3
A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve.
A2
1
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Putnam 1956 A2
Prove that every positive integer has a multiple whose decimal representation involves all ten digits.
A1
1
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Putnam 1956 A1
Evaluate \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x}, where
a
>
0
a>0
a
>
0
and
a
≠
1.
a\ne 1.
a
=
1.
B5
1
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combi proof without induction
Show that a graph with 2n points and
n
2
+
1
n^2 + 1
n
2
+
1
edges necessarily contains a 3-cycle, but that we can find a graph with 2n points and
n
2
n^2
n
2
edges without a 3-cycle.please prove it without induction .