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Putnam
1969 Putnam
1969 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1969 B6
Let
A
A
A
and
B
B
B
be matrices of size
3
×
2
3\times 2
3
×
2
and
2
×
3
2\times 3
2
×
3
respectively. Suppose that
A
B
=
(
8
2
−
2
2
5
4
−
2
4
5
)
.
AB =\begin{pmatrix} 8 & 2 & -2\\ 2 & 5 &4 \\ -2 &4 &5 \end{pmatrix}.
A
B
=
8
2
−
2
2
5
4
−
2
4
5
.
Show that the product
B
A
BA
B
A
is equal to
(
9
0
0
9
)
.
\begin{pmatrix} 9 &0\\ 0 &9 \end{pmatrix}.
(
9
0
0
9
)
.
B5
1
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Putnam 1969 B5
Let
a
1
<
a
2
<
…
a_1 <a_2 < \ldots
a
1
<
a
2
<
…
be an increasing sequence of positive integers. Let the series
∑
i
=
1
∞
1
a
i
\sum_{i=1}^{\infty} \frac{1}{a_i }
i
=
1
∑
∞
a
i
1
be convergent. For any real number
x
x
x
, let
k
(
x
)
k(x)
k
(
x
)
be the number of the
a
i
a_i
a
i
which do not exceed
x
x
x
. Show that
lim
x
→
∞
k
(
x
)
x
=
0.
\lim_{x\to \infty} \frac{k(x)}{x}=0.
lim
x
→
∞
x
k
(
x
)
=
0.
B4
1
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Putnam 1969 B4
Show that any curve of unit length can be covered by a closed rectangle of area 1 \slash 4.
B3
1
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Putnam 1969 B3
The terms of a sequence
(
T
n
)
(T_n)
(
T
n
)
satisfy
T
n
T
n
+
1
=
n
T_n T_{n+1} =n
T
n
T
n
+
1
=
n
for all positive integers
n
n
n
and
lim
n
→
∞
T
n
T
n
+
1
=
1.
\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.
n
→
∞
lim
T
n
+
1
T
n
=
1.
Show that
π
T
1
2
=
2.
\pi T_{1}^{2}=2.
π
T
1
2
=
2.
B2
1
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Putnam 1969 B2
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if "two' is replaced by "three'?
A6
1
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Putnam 1969 A6
Let a sequence
(
x
n
)
(x_n)
(
x
n
)
be given and let
y
n
=
x
n
−
1
+
2
x
n
y_n = x_{n-1} +2 x_n
y
n
=
x
n
−
1
+
2
x
n
for
n
>
1.
n>1.
n
>
1.
Suppose that the sequence
(
y
n
)
(y_n)
(
y
n
)
converges. Prove that the sequence
(
x
n
)
(x_n)
(
x
n
)
converges, too.
A5
1
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Putnam 1969 A5
Let
u
(
t
)
u(t)
u
(
t
)
be a continuous function in the system of differential equations
d
x
d
t
=
−
2
y
+
u
(
t
)
,
d
y
d
t
=
−
2
x
+
u
(
t
)
.
\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).
d
t
d
x
=
−
2
y
+
u
(
t
)
,
d
t
d
y
=
−
2
x
+
u
(
t
)
.
Show that, regardless of the choice of
u
(
t
)
u(t)
u
(
t
)
, the solution of the system which satisfies
x
=
x
0
,
y
=
y
0
x=x_0 , y=y_0
x
=
x
0
,
y
=
y
0
at
t
=
0
t=0
t
=
0
will never pass through
(
0
,
0
)
(0, 0)
(
0
,
0
)
unless
x
0
=
y
0
.
x_0 =y_0.
x
0
=
y
0
.
When
x
0
=
y
0
x_0 =y_0
x
0
=
y
0
, show that, for any positive value
t
0
t_0
t
0
of
t
t
t
, it is possible to choose
u
(
t
)
u(t)
u
(
t
)
so the solution is equal to
(
0
,
0
)
(0,0)
(
0
,
0
)
when
t
=
t
0
.
t=t_0 .
t
=
t
0
.
A4
1
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Putnam 1969 A4
Show that
∫
0
1
x
x
d
x
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
n
.
\int_{0}^{1} x^{x} \, dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^n }.
∫
0
1
x
x
d
x
=
n
=
1
∑
∞
n
n
(
−
1
)
n
+
1
.
A3
1
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Putnam 1969 A3
Let
P
P
P
be a non-selfintersecting closed polygon with
n
n
n
sides. Let its vertices be
P
1
,
P
2
,
…
,
P
n
.
P_1 , P_2 ,\ldots, P_n .
P
1
,
P
2
,
…
,
P
n
.
Let
m
m
m
other points,
Q
1
,
Q
2
,
…
,
Q
m
Q_1 , Q_2 ,\ldots, Q_m
Q
1
,
Q
2
,
…
,
Q
m
, interior to
P
P
P
, be given. Let the figure be triangulated. This means that certain pairs of the
(
n
+
m
)
(n+m)
(
n
+
m
)
points
P
1
,
…
,
Q
m
P_1 ,\ldots , Q_m
P
1
,
…
,
Q
m
are connected by line segments such that (i) the resulting figure consists exclusively of a set
T
T
T
of triangles, (ii) if two different triangles in
T
T
T
have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in
T
T
T
is precisely the set of the
(
n
+
m
)
(n+m)
(
n
+
m
)
points
P
1
,
…
,
Q
m
.
P_1 ,\ldots , Q_m.
P
1
,
…
,
Q
m
.
How many triangles are in
T
T
T
?
A2
1
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Putnam 1969 A2
Let
D
n
D_n
D
n
be the determinant of order
n
n
n
of which the element in the
i
i
i
-th row and the
j
j
j
-th column is
∣
i
−
j
∣
.
|i-j|.
∣
i
−
j
∣.
Show that
D
n
D_n
D
n
is equal to
(
−
1
)
n
−
1
(
n
−
1
)
2
n
−
2
.
(-1)^{n-1}(n-1)2^{n-2}.
(
−
1
)
n
−
1
(
n
−
1
)
2
n
−
2
.
A1
1
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Putnam 1969 A1
Let
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
be a polynomial with real coefficients in the real variables
x
x
x
and
y
y
y
defined over the entire
x
y
xy
x
y
-plane. What are the possibilities for the range of
f
(
x
,
y
)
?
f(x,y)?
f
(
x
,
y
)?
B1
1
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Sum of factors
Let
n
n
n
be a positive integer such that
24
∣
n
+
1
24\mid n+1
24
∣
n
+
1
. Prove that the sum of the positive divisors of
n
n
n
is divisble by 24.