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Putnam
1962 Putnam
1962 Putnam
Part of
Putnam
Subcontests
(12)
A5
1
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Putnam 1962 A5
Evaluate
∑
k
=
0
n
(
n
k
)
k
2
.
\sum_{k=0}^{n} \binom{n}{k}k^{2}.
k
=
0
∑
n
(
k
n
)
k
2
.
B6
1
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Putnam 1962 B6
Let
f
(
x
)
=
∑
k
=
0
n
a
k
sin
k
x
+
b
k
cos
k
x
,
f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,
f
(
x
)
=
k
=
0
∑
n
a
k
sin
k
x
+
b
k
cos
k
x
,
where
a
k
a_k
a
k
and
b
k
b_k
b
k
are constants. Show that if
∣
f
(
x
)
∣
≤
1
|f(x)| \leq 1
∣
f
(
x
)
∣
≤
1
for
x
∈
[
0
,
2
π
]
x \in [0, 2 \pi]
x
∈
[
0
,
2
π
]
and there exist
0
≤
x
1
<
x
2
<
…
<
x
2
n
<
2
π
0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi
0
≤
x
1
<
x
2
<
…
<
x
2
n
<
2
π
with
∣
f
(
x
i
)
∣
=
1
,
|f(x_i )|=1,
∣
f
(
x
i
)
∣
=
1
,
then
f
(
x
)
=
cos
(
n
x
+
a
)
f(x)= \cos(nx +a)
f
(
x
)
=
cos
(
n
x
+
a
)
for some constant
a
.
a.
a
.
B5
1
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Putnam 1962 B5
Prove that for every integer
n
n
n
greater than
1
:
1:
1
:
3
n
+
1
2
n
+
2
<
(
1
n
)
n
+
(
2
n
)
n
+
…
+
(
n
n
)
n
<
2.
\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.
2
n
+
2
3
n
+
1
<
(
n
1
)
n
+
(
n
2
)
n
+
…
+
(
n
n
)
n
<
2.
B4
1
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Putnam 1962 B4
The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.
B3
1
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Putnam 1962 B3
Let
S
S
S
be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside
S
S
S
. Prove that
S
S
S
is bounded.
B2
1
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Putnam 1962 B2
Let
S
S
S
be the set of all subsets of the positive integers. Construct a function
f
:
R
→
S
f \colon \mathbb{R} \rightarrow S
f
:
R
→
S
such that
f
(
a
)
f(a)
f
(
a
)
is a proper subset of
f
(
b
)
f(b)
f
(
b
)
whenever
a
<
b
.
a <b.
a
<
b
.
B1
1
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Putnam 1962 B1
Let
x
(
n
)
=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
x^{(n)}=x(x-1)\cdots (x-n+1)
x
(
n
)
=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
for
n
n
n
a positive integer and let
x
(
0
)
=
1.
x^{(0)}=1.
x
(
0
)
=
1.
Prove that
(
x
+
y
)
(
n
)
=
∑
k
=
0
n
(
n
k
)
x
(
k
)
y
(
n
−
k
)
.
(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.
(
x
+
y
)
(
n
)
=
k
=
0
∑
n
(
k
n
)
x
(
k
)
y
(
n
−
k
)
.
A6
1
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Putnam 1962 A6
Let
S
S
S
be a set of rational numbers such that whenever
a
a
a
and
b
b
b
are members of
S
S
S
, so are
a
b
ab
ab
and
a
+
b
a+b
a
+
b
, and having the property that for every rational number
r
r
r
exactly one of the following three statements is true:
r
∈
S
,
−
r
∈
S
,
r
=
0.
r\in S,\;\; -r\in S,\;\;r =0.
r
∈
S
,
−
r
∈
S
,
r
=
0.
Prove that
S
S
S
is the set of all positive rational numbers.
A4
1
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Putnam 1962 A4
Assume that
∣
f
(
x
)
∣
≤
1
|f(x)|\leq 1
∣
f
(
x
)
∣
≤
1
and
∣
f
′
′
(
x
)
∣
≤
1
|f''(x)|\leq 1
∣
f
′′
(
x
)
∣
≤
1
for all
x
x
x
on an interval of length at least
2.
2.
2.
Show that
∣
f
′
(
x
)
∣
≤
2
|f'(x)|\leq 2
∣
f
′
(
x
)
∣
≤
2
on the interval.
A3
1
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Putnam 1962 A3
In a triangle
A
B
C
ABC
A
BC
, let
A
′
A'
A
′
be a point on the segment
B
C
BC
BC
,
B
′
B'
B
′
be a point on the segment
C
A
CA
C
A
and
C
′
C'
C
′
a point on the segment
A
B
AB
A
B
such that
A
B
′
B
′
C
=
B
C
′
C
′
A
=
C
A
′
A
′
B
=
k
,
\frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,
B
′
C
A
B
′
=
C
′
A
B
C
′
=
A
′
B
C
A
′
=
k
,
where
k
k
k
is a positive constant. Let
△
\triangle
△
be the triangle formed by the interesctions of
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
. Prove that the areas of
△
\triangle
△
and
A
B
C
ABC
A
BC
are in the ratio
(
k
−
1
)
2
k
2
+
k
+
1
.
\frac{(k-1)^{2}}{k^2 +k+1}.
k
2
+
k
+
1
(
k
−
1
)
2
.
A2
1
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Putnam 1962 A2
Find every real-valued function
f
f
f
whose domain is an interval
I
I
I
(finite or infinite) having
0
0
0
as a left-hand endpoint, such that for every positive
x
∈
I
x\in I
x
∈
I
the average of
f
f
f
over the closed interval
[
0
,
x
]
[0,x]
[
0
,
x
]
is equal to
f
(
0
)
f
(
x
)
.
\sqrt{ f(0) f(x)}.
f
(
0
)
f
(
x
)
.
A1
1
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convex quadr. exists given 5 points, no 3 collinear 2010 Chile Ibero TST
Consider
5
5
5
points in the plane, such that there are no
3
3
3
of them collinear. Prove that there is a convex quadrilateral with vertices at
4
4
4
points.