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Putnam
1973 Putnam
1973 Putnam
Part of
Putnam
Subcontests
(12)
B1
1
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Putnam 1973 B1
Let
a
1
,
a
2
,
…
a
2
n
+
1
a_1, a_2, \ldots a_{2n+1}
a
1
,
a
2
,
…
a
2
n
+
1
be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of
n
n
n
integers with equal sums. Prove
a
1
=
a
2
=
⋯
=
a
2
n
+
1
.
a_{1}=a_2 =\cdots=a_{2n+1}.
a
1
=
a
2
=
⋯
=
a
2
n
+
1
.
B6
1
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Putnam 1973 B6
On the domain
0
≤
θ
≤
2
π
:
0\leq \theta \leq 2\pi:
0
≤
θ
≤
2
π
:
(a) Prove that
sin
2
θ
⋅
sin
2
θ
\sin^{2}\theta \cdot \sin 2\theta
sin
2
θ
⋅
sin
2
θ
takes its maximum at
π
3
\frac{\pi}{3}
3
π
and
4
π
3
\frac{4 \pi}{3}
3
4
π
(and hence its minimum at
2
π
3
\frac{2 \pi}{3}
3
2
π
and
5
π
3
\frac{ 5 \pi}{3}
3
5
π
). (b) Show that
∣
sin
2
θ
⋅
sin
3
2
θ
⋅
sin
3
4
θ
⋯
sin
3
2
n
−
1
θ
⋅
sin
2
n
θ
∣
| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |
∣
sin
2
θ
⋅
sin
3
2
θ
⋅
sin
3
4
θ
⋯
sin
3
2
n
−
1
θ
⋅
sin
2
n
θ
∣
takes its maximum at
4
π
3
\frac{4 \pi}{3}
3
4
π
(the maximum may also be attained at other points). (c) Derive the inequality:
sin
2
θ
⋅
sin
2
2
θ
⋯
sin
2
2
n
θ
≤
(
3
4
)
n
.
\sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.
sin
2
θ
⋅
sin
2
2
θ
⋯
sin
2
2
n
θ
≤
(
4
3
)
n
.
B5
1
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Putnam 1973 B5
(a) Let
z
z
z
be a solution of the quadratic equation
a
z
2
+
b
z
+
c
=
0
az^2 +bz+c=0
a
z
2
+
b
z
+
c
=
0
and let
n
n
n
be a positive integer. Show that
z
z
z
can be expressed as a rational function of
z
n
,
a
,
b
,
c
.
z^n , a,b,c.
z
n
,
a
,
b
,
c
.
(b) Using (a) or by any other means, express
x
x
x
as a rational function of
x
3
x^{3}
x
3
and
x
+
1
x
.
x+\frac{1}{x}.
x
+
x
1
.
B4
1
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Putnam 1973 B4
(a) On
[
0
,
1
]
[0, 1]
[
0
,
1
]
, let
f
(
x
)
f(x)
f
(
x
)
have a continuous derivative satisfying
0
<
f
′
(
x
)
≤
1
0 <f'(x) \leq1
0
<
f
′
(
x
)
≤
1
. Also suppose that
f
(
0
)
=
0.
f(0) = 0.
f
(
0
)
=
0.
Prove that
(
∫
0
1
f
(
x
)
d
x
)
2
≥
∫
0
1
f
(
x
)
3
d
x
.
\left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.
(
∫
0
1
f
(
x
)
d
x
)
2
≥
∫
0
1
f
(
x
)
3
d
x
.
(b) Show an example in which equality occurs.
B3
1
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Putnam 1973 B3
Consider an integer
p
>
1
p>1
p
>
1
with the property that the polynomial
x
2
−
x
+
p
x^2 - x + p
x
2
−
x
+
p
takes prime values for all integers
x
x
x
such that
0
≤
x
<
p
0\leq x <p
0
≤
x
<
p
. Show that there is exactly one triple of integers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying the conditions:
b
2
−
4
a
c
=
1
−
4
p
,
0
<
a
≤
c
,
−
a
≤
b
<
a
.
b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.
b
2
−
4
a
c
=
1
−
4
p
,
0
<
a
≤
c
,
−
a
≤
b
<
a
.
B2
1
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Putnam 1973 B2
Let
z
=
x
+
y
i
z=x+yi
z
=
x
+
y
i
be a complex number with
x
x
x
and
y
y
y
rational and with
∣
z
∣
=
1.
|z|=1.
∣
z
∣
=
1.
Prove that the number
∣
z
2
n
−
1
∣
|z^{2n} -1|
∣
z
2
n
−
1∣
is rational for every integer
n
n
n
.
A6
1
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Putnam 1973 A6
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.
A5
1
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Putnam 1973 A5
A particle moves in
3
3
3
-space according to the equations:
d
x
d
t
=
y
z
,
d
y
d
t
=
x
z
,
d
z
d
t
=
x
y
.
\frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy.
d
t
d
x
=
yz
,
d
t
d
y
=
x
z
,
d
t
d
z
=
x
y
.
Show that: (a) If two of
x
(
0
)
,
y
(
0
)
,
z
(
0
)
x(0), y(0), z(0)
x
(
0
)
,
y
(
0
)
,
z
(
0
)
equal
0
,
0,
0
,
then the particle never moves. (b) If
x
(
0
)
=
y
(
0
)
=
1
,
z
(
0
)
=
0
,
x(0)=y(0)=1, z(0)=0,
x
(
0
)
=
y
(
0
)
=
1
,
z
(
0
)
=
0
,
then the solution is
x
(
t
)
=
sec
t
,
y
(
t
)
=
sec
t
,
z
(
t
)
=
tan
t
;
x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t;
x
(
t
)
=
sec
t
,
y
(
t
)
=
sec
t
,
z
(
t
)
=
tan
t
;
whereas if
x
(
0
)
=
y
(
0
)
=
1
,
z
(
0
)
=
−
1
,
x(0)=y(0)=1, z(0)=-1,
x
(
0
)
=
y
(
0
)
=
1
,
z
(
0
)
=
−
1
,
then
x
(
t
)
=
1
t
+
1
,
y
(
t
)
=
1
t
+
1
,
z
(
t
)
=
−
1
t
+
1
.
x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}.
x
(
t
)
=
t
+
1
1
,
y
(
t
)
=
t
+
1
1
,
z
(
t
)
=
−
t
+
1
1
.
(c) If at least two of the values
x
(
0
)
,
y
(
0
)
,
z
(
0
)
x(0), y(0), z(0)
x
(
0
)
,
y
(
0
)
,
z
(
0
)
are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past (a point
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
in
3
3
3
-space "moves to infinity" if its distance from the origin approaches infinity).
A4
1
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Putnam 1973 A4
How many zeroes does the function
f
(
x
)
=
2
x
−
1
−
x
2
f(x)=2^x -1 -x^2
f
(
x
)
=
2
x
−
1
−
x
2
have on the real line?
A3
1
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Putnam 1973 A3
Let
n
n
n
be a fixed positive integer and let
b
(
n
)
b(n)
b
(
n
)
be the minimum value of
k
+
n
k
,
k+\frac{n}{k},
k
+
k
n
,
where
k
k
k
is allowed to range through all positive integers. Prove that
⌊
b
(
n
)
⌋
=
⌊
4
n
+
1
⌋
.
\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.
⌊
b
(
n
)⌋
=
⌊
4
n
+
1
⌋
.
A2
1
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Putnam 1973 A2
Consider an infinite series whose
n
n
n
-th term is \pm (1\slash n), the
±
\pm
±
signs being determined according to a pattern that repeats periodically in blocks of eight (there are
2
8
2^{8}
2
8
possible patterns).(a) Show that a sufficient condition for the series to be conditionally convergent is that there are four "
+
+
+
" signs and four "
−
-
−
" signs in the block of eight signs. (b) Is this sufficient condition also necessary?
A1
1
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Putnam 1973 A1
(a) Let
A
B
C
ABC
A
BC
be any triangle. Let
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
be points on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Suppose that
B
X
≤
X
C
,
C
Y
≤
Y
A
,
A
Z
≤
Z
B
BX \leq XC, CY \leq YA, AZ \leq ZB
BX
≤
XC
,
C
Y
≤
Y
A
,
A
Z
≤
ZB
. Show that the area of the triangle
X
Y
Z
XYZ
X
Y
Z
\geq 1\slash 4 times the area of
A
B
C
.
ABC.
A
BC
.
(b) Let
A
B
C
ABC
A
BC
be any triangle, and let
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
be points on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Using (a) or by any other method, show: One of the three corner triangles
A
Z
Y
,
B
X
Z
,
C
Y
X
AZY, BXZ, CYX
A
Z
Y
,
BXZ
,
C
Y
X
has an area
≤
\leq
≤
area of the triangle
X
Y
Z
.
XYZ.
X
Y
Z
.