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Putnam
1980 Putnam
1980 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1980 B6
An infinite array of rational numbers
G
(
d
,
n
)
G(d, n)
G
(
d
,
n
)
is defined for integers
d
d
d
and
n
n
n
with
1
≤
d
≤
n
1\leq d \leq n
1
≤
d
≤
n
as follows:
G
(
1
,
n
)
=
1
n
,
G
(
d
,
n
)
=
d
n
∑
i
=
d
n
G
(
d
−
1
,
i
−
1
)
for
d
>
1.
G(1, n)= \frac{1}{n}, \;\;\; G(d,n)= \frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \; \text{for} \; d>1.
G
(
1
,
n
)
=
n
1
,
G
(
d
,
n
)
=
n
d
i
=
d
∑
n
G
(
d
−
1
,
i
−
1
)
for
d
>
1.
For
1
<
d
<
p
1 < d < p
1
<
d
<
p
and
p
p
p
prime, prove that
G
(
d
,
p
)
G(d, p)
G
(
d
,
p
)
is expressible as a quotient s\slash t of integers
s
s
s
and
t
t
t
with
t
t
t
not divisible by
p
.
p.
p
.
B5
1
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Putnam 1980 B5
For each
t
≥
0
t \geq 0
t
≥
0
let
S
t
S_t
S
t
be the set of all nonnegative, increasing, convex, continuous, real-valued functions
f
(
x
)
f(x)
f
(
x
)
defined on the closed interval
[
0
,
1
]
[0,1]
[
0
,
1
]
for which f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)). Define necessary and sufficient conditions on
t
t
t
for
S
t
S_t
S
t
to be closed under multiplication.
B4
1
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Putnam 1980 B4
Let
A
1
,
A
2
,
…
,
A
1066
A_1 , A_2 ,\ldots, A_{1066}
A
1
,
A
2
,
…
,
A
1066
be subsets of a finite set
X
X
X
such that
∣
A
i
∣
>
1
2
∣
X
∣
|A_i | > \frac{1}{2} |X|
∣
A
i
∣
>
2
1
∣
X
∣
for
1
≤
i
≤
1066.
1\leq i \leq 1066.
1
≤
i
≤
1066.
Prove that there exist ten elements
x
1
,
x
2
,
…
,
x
10
x_1 ,x_2 ,\ldots , x_{10}
x
1
,
x
2
,
…
,
x
10
of
X
X
X
such that every
A
i
A_i
A
i
contains at least one of
x
1
,
x
2
,
…
,
x
10
.
x_1 , x_2 ,\ldots, x_{10}.
x
1
,
x
2
,
…
,
x
10
.
B3
1
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Putnam 1980 B3
For which real numbers
a
a
a
does the sequence
(
u
n
)
(u_n )
(
u
n
)
defined by the initial condition
u
0
=
a
u_0 =a
u
0
=
a
and the recursion
u
n
+
1
=
2
u
n
−
n
2
u_{n+1} =2u_n - n^2
u
n
+
1
=
2
u
n
−
n
2
have
u
n
>
0
u_n >0
u
n
>
0
for all
n
≥
0
?
n \geq 0?
n
≥
0
?
B2
1
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Putnam 1980 B2
Let
S
S
S
be the solid in three-dimensional space consisting of all points
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
satisfying the following six simultaneous conditions:
x
,
y
,
z
≥
0
,
x
+
y
+
z
≤
11
,
2
x
+
4
y
+
3
z
≤
36
,
2
x
+
3
z
≤
44.
x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.
x
,
y
,
z
≥
0
,
x
+
y
+
z
≤
11
,
2
x
+
4
y
+
3
z
≤
36
,
2
x
+
3
z
≤
44.
a) Determine the number
V
V
V
of vertices of
S
.
S.
S
.
b) Determine the number
E
E
E
of edges of
S
.
S.
S
.
c) Sketch in the
b
c
bc
b
c
-plane the set of points
(
b
,
c
)
(b, c)
(
b
,
c
)
such that
(
2
,
5
,
4
)
(2,5,4)
(
2
,
5
,
4
)
is one of the points
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
at which the linear function
b
x
+
c
y
+
z
bx + cy + z
b
x
+
cy
+
z
assumes its maximum value on
S
.
S.
S
.
B1
1
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Putnam 1980 B1
For which real numbers
c
c
c
is
e
x
+
e
−
x
2
≤
e
c
x
2
\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }
2
e
x
+
e
−
x
≤
e
c
x
2
for all real
x
?
x?
x
?
A6
1
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Putnam 1980 A6
Let
C
C
C
be the class of all real valued continuously differentiable functions
f
f
f
on the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
with
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
1
)
=
1.
f(1)=1 .
f
(
1
)
=
1.
Determine the largest real number
u
u
u
such that
u
≤
∫
0
1
∣
f
′
(
x
)
−
f
(
x
)
∣
d
x
u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx
u
≤
∫
0
1
∣
f
′
(
x
)
−
f
(
x
)
∣
d
x
for all
f
f
f
in
C
.
C.
C
.
A5
1
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Putnam 1980 A5
Let
P
(
t
)
P(t)
P
(
t
)
be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations
∫
0
x
P
(
t
)
sin
t
d
t
=
0
,
∫
0
x
P
(
t
)
cos
t
d
t
=
0
\int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0
∫
0
x
P
(
t
)
sin
t
d
t
=
0
,
∫
0
x
P
(
t
)
cos
t
d
t
=
0
has only finitely many solutions
x
.
x.
x
.
A4
1
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Putnam 1980 A4
a) Prove that there exist integers
a
,
b
,
c
a, b, c
a
,
b
,
c
not all zero and each of absolute value less than one million, such that
∣
a
+
b
2
+
c
3
∣
<
1
0
−
11
.
|a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .
∣
a
+
b
2
+
c
3
∣
<
1
0
−
11
.
b) Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be integers, not all zero and each of absolute value less than one million. Prove that
∣
a
+
b
2
+
c
3
∣
>
1
0
−
21
.
|a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .
∣
a
+
b
2
+
c
3
∣
>
1
0
−
21
.
A3
1
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Putnam 1980 A3
Evaluate \int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.
A2
1
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Putnam 1980 A2
Let
r
r
r
and
s
s
s
be positive integers. Derive a formula for the number of ordered quadruples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
of positive integers such that
3
r
⋅
7
s
=
lcm
(
a
,
b
,
c
)
=
lcm
(
a
,
b
,
d
)
=
lcm
(
a
,
c
,
d
)
=
lcm
(
b
,
c
,
d
)
,
3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),
3
r
⋅
7
s
=
lcm
(
a
,
b
,
c
)
=
lcm
(
a
,
b
,
d
)
=
lcm
(
a
,
c
,
d
)
=
lcm
(
b
,
c
,
d
)
,
depending only on
r
r
r
and
s
.
s.
s
.
A1
1
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Putnam 1980 A1
Let
b
b
b
and
c
c
c
be fixed real numbers and let the ten points
(
j
,
y
j
)
(j,y_j )
(
j
,
y
j
)
for
j
=
1
,
2
,
…
,
10
j=1,2,\ldots,10
j
=
1
,
2
,
…
,
10
lie on the parabola
y
=
x
2
+
b
x
+
c
.
y =x^2 +bx+c.
y
=
x
2
+
b
x
+
c
.
For
j
=
1
,
2
,
…
,
9
j=1,2,\ldots, 9
j
=
1
,
2
,
…
,
9
let
I
j
I_j
I
j
be the intersection of the tangents to the given parabola at
(
j
,
y
j
)
(j, y_j )
(
j
,
y
j
)
and
(
j
+
1
,
y
j
+
1
)
.
(j+1, y_{j+1}).
(
j
+
1
,
y
j
+
1
)
.
Determine the poynomial function
y
=
g
(
x
)
y=g(x)
y
=
g
(
x
)
of least degree whose graph passes through all nine points
I
j
.
I_j .
I
j
.