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Putnam
1940 Putnam
1940 Putnam
Part of
Putnam
Subcontests
(15)
A1
1
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Putnam 1940 A1
Prove that if
f
(
x
)
f(x)
f
(
x
)
is a polynomial with integer coefficients and there exists an integer
k
k
k
such that none of
f
(
1
)
,
…
,
f
(
k
)
f(1),\ldots,f(k)
f
(
1
)
,
…
,
f
(
k
)
is divisible by
k
k
k
, then
f
(
x
)
f(x)
f
(
x
)
has no integral root.
B7
1
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Putnam 1940 B7
Which is greater
n
n
+
1
or
n
+
1
n
\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}
n
n
+
1
or
n
+
1
n
where
n
>
8
?
n>8?
n
>
8
?
B6
1
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Putnam 1940 B6
Prove that the determinant of the matrix
(
a
1
2
+
k
a
1
a
2
a
1
a
3
…
a
1
a
n
a
2
a
1
a
2
2
+
k
a
2
a
3
…
a
2
a
n
…
…
…
…
…
a
n
a
1
a
n
a
2
a
n
a
3
…
a
n
2
+
k
)
\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\ a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}
a
1
2
+
k
a
2
a
1
…
a
n
a
1
a
1
a
2
a
2
2
+
k
…
a
n
a
2
a
1
a
3
a
2
a
3
…
a
n
a
3
…
…
…
…
a
1
a
n
a
2
a
n
…
a
n
2
+
k
is divisible by
k
n
−
1
k^{n-1}
k
n
−
1
and find its other factor.
B4
1
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Putnam 1940 B4
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
a
x
2
+
b
y
2
+
c
z
2
=
1
(
where
a
b
c
≠
0
)
ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)
a
x
2
+
b
y
2
+
c
z
2
=
1
(
where
ab
c
=
0
)
is the sphere
x
2
+
y
2
+
z
2
=
1
a
+
1
b
+
1
c
.
x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.
x
2
+
y
2
+
z
2
=
a
1
+
b
1
+
c
1
.
B3
1
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Putnam 1940 B3
Let
p
>
0
p>0
p
>
0
be a real constant. From any point
(
a
,
b
)
(a,b)
(
a
,
b
)
in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola
y
2
=
4
p
x
y^2=4px
y
2
=
4
p
x
. ii) These are real and distinct if
4
(
2
−
p
)
3
+
27
p
b
2
<
0
4(2-p)^3 +27pb^2<0
4
(
2
−
p
)
3
+
27
p
b
2
<
0
. iii) Two of them coincide if
(
a
,
b
)
(a,b)
(
a
,
b
)
lies on the curve
27
p
y
2
=
4
(
x
−
2
p
)
3
27py^2=4(x-2p)^3
27
p
y
2
=
4
(
x
−
2
p
)
3
. iv) All three coincide only if
a
=
2
p
a=2p
a
=
2
p
and
b
=
0
b=0
b
=
0
.
B2
1
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Putnam 1940 B2
A cylindrical hole of radius
r
r
r
is bored through a cylinder of radiues
R
R
R
(
r
≤
R
r\leq R
r
≤
R
) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form
S
=
8
r
2
∫
0
1
1
−
v
2
(
1
−
v
2
)
(
1
−
m
2
v
2
)
d
v
,
where
m
=
r
R
.
S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.
S
=
8
r
2
∫
0
1
(
1
−
v
2
)
(
1
−
m
2
v
2
)
1
−
v
2
d
v
,
where
m
=
R
r
.
ii) If
K
=
∫
0
1
1
(
1
−
v
2
)
(
1
−
m
2
v
2
)
d
v
K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv
K
=
∫
0
1
(
1
−
v
2
)
(
1
−
m
2
v
2
)
1
d
v
and
E
=
∫
0
1
1
−
m
2
v
2
1
−
v
2
d
v
E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv
E
=
∫
0
1
1
−
v
2
1
−
m
2
v
2
d
v
. show that
S
=
8
[
R
2
E
−
(
R
2
−
r
2
)
K
]
.
S=8[R^2 E - (R^2 - r^2 )K].
S
=
8
[
R
2
E
−
(
R
2
−
r
2
)
K
]
.
B1
1
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Putnam 1940 B1
A projectile, thrown with initial velocity
v
0
v_0
v
0
in a direction making angle
α
\alpha
α
with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when
sin
α
log
(
sec
α
+
tan
α
)
=
1.
\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.
sin
α
lo
g
(
sec
α
+
tan
α
)
=
1.
A8
1
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Putnam 1940 A8
A triangle is bounded by the lines
a
1
x
+
b
1
y
+
c
1
=
0
a_1 x+ b_1 y +c_1=0
a
1
x
+
b
1
y
+
c
1
=
0
,
a
2
x
+
b
2
y
+
c
2
=
0
a_2 x+ b_2 y +c_2=0
a
2
x
+
b
2
y
+
c
2
=
0
and
a
2
x
+
b
2
y
+
c
2
=
0
a_2 x+ b_2 y +c_2=0
a
2
x
+
b
2
y
+
c
2
=
0
. Show that its area, disregarding sign, is
Δ
2
2
(
a
2
b
3
−
a
3
b
2
)
(
a
3
b
1
−
a
1
b
3
)
(
a
1
b
2
−
a
2
b
1
)
,
\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},
2
(
a
2
b
3
−
a
3
b
2
)
(
a
3
b
1
−
a
1
b
3
)
(
a
1
b
2
−
a
2
b
1
)
Δ
2
,
where
Δ
\Delta
Δ
is the discriminant of the matrix
M
=
(
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
)
.
M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.
M
=
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3
.
A7
1
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Putnam 1940 A7
If
∑
i
=
1
∞
u
i
2
\sum_{i=1}^{\infty} u_{i}^{2}
∑
i
=
1
∞
u
i
2
and
∑
i
=
1
∞
v
i
2
\sum_{i=1}^{\infty} v_{i}^{2}
∑
i
=
1
∞
v
i
2
are convergent series of real numbers, prove that
∑
i
=
1
∞
(
u
i
−
v
i
)
p
\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}
i
=
1
∑
∞
(
u
i
−
v
i
)
p
is convergent, where
p
≥
2
p\geq 2
p
≥
2
is an integer.
A6
1
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Putnam 1940 A6
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial of degree
n
n
n
such that
f
(
x
)
p
f(x)^{p}
f
(
x
)
p
is divisible by
f
′
(
x
)
q
f'(x)^{q}
f
′
(
x
)
q
for some positive integers
p
,
q
p,q
p
,
q
. Prove that
f
(
x
)
f(x)
f
(
x
)
is divisible by
f
′
(
x
)
f'(x)
f
′
(
x
)
and that
f
(
x
)
f(x)
f
(
x
)
has a single root of multiplicity
n
n
n
.
A5
1
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Putnam 1940 A5
Prove that the simultaneous equations
x
4
−
x
2
=
y
4
−
y
2
=
z
4
−
z
2
x^4 -x^2 =y^4 -y^2 =z^4 -z^2
x
4
−
x
2
=
y
4
−
y
2
=
z
4
−
z
2
are satisfied by the points of
4
4
4
straight lines and
6
6
6
ellipses, and by no other points.
A4
1
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Putnam 1940 A4
Let
p
p
p
be a real constant. The parabola
y
2
=
−
4
p
x
y^2=-4px
y
2
=
−
4
p
x
rolls without slipping around the parabola
y
2
=
4
p
x
y^2=4px
y
2
=
4
p
x
. Find the equation of the locus of the vertex of the rolling parabola.
A3
1
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Putnam 1940 A3
Let
a
a
a
be a real number. Find all real-valued functions
f
f
f
such that
∫
f
(
x
)
a
d
x
=
(
∫
f
(
x
)
d
x
)
a
\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}
∫
f
(
x
)
a
d
x
=
(
∫
f
(
x
)
d
x
)
a
when constants of integration are suitably chosen.
A2
1
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Putnam 1940 A2
Let
A
,
B
A,B
A
,
B
be two fixed points on the curve
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
,
f
f
f
is continuous with continuous derivative and the arc
A
B
^
\widehat{AB}
A
B
is concave to the chord
A
B
AB
A
B
. If
P
P
P
is a point on the arc
A
B
^
\widehat{AB}
A
B
for which
A
P
+
P
B
AP+PB
A
P
+
PB
is maximal, prove that
P
A
PA
P
A
and
P
B
PB
PB
are equally inclined to the tangent to the curve
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
at
P
P
P
.
B5
1
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a, b,c rational roots of x^3+ax^2 + bx + c = 0
Suppose that the rational numbers
a
,
b
a, b
a
,
b
and
c
c
c
are the roots of the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3+ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
. Find all such rational numbers
a
,
b
a, b
a
,
b
and
c
c
c
. Justify your answer