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Putnam
1965 Putnam
1965 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1965 B6
If
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
are four distinct points such that every circle through
A
A
A
and
B
B
B
intersects (or coincides with) every circle through
C
C
C
and
D
D
D
, prove that the four points are either collinear (all on one line) or concyclic (all on one circle).
B5
1
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Putnam 1965 B5
Consider collections of unordered pairs of
V
V
V
different objects
a
a
a
,
b
b
b
,
c
c
c
,
…
\ldots
…
,
k
k
k
. Three pairs such as
a
b
ab
ab
,
b
c
bc
b
c
,
a
b
ab
ab
are said to form a triangle. Prove that, if
4
E
≤
V
2
4E\leq V^2
4
E
≤
V
2
, it is possible to choose
E
E
E
pairs so that no triangle is formed.
B4
1
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Putnam 1965 B4
Consider the function
f
(
x
,
n
)
=
(
n
0
)
+
(
n
2
)
x
+
(
n
4
)
x
2
+
⋯
(
n
1
)
+
(
n
3
)
x
+
(
n
5
)
x
2
+
⋯
,
f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots},
f
(
x
,
n
)
=
(
1
n
)
+
(
3
n
)
x
+
(
5
n
)
x
2
+
⋯
(
0
n
)
+
(
2
n
)
x
+
(
4
n
)
x
2
+
⋯
,
where
n
n
n
is a positive integer. Express
f
(
x
,
n
+
1
)
f(x,n+1)
f
(
x
,
n
+
1
)
rationally in terms of
f
(
x
,
n
)
f(x,n)
f
(
x
,
n
)
and
x
x
x
. Hence, or otherwise, evaluate
lim
n
→
∞
f
(
x
,
n
)
\textstyle\lim_{n\to\infty}f(x,n)
lim
n
→
∞
f
(
x
,
n
)
for suitable fixed values of
x
x
x
. (The symbols
(
n
r
)
\textstyle\binom nr
(
r
n
)
represent the binomial coefficients.)
B3
1
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Putnam 1965 B3
Prove that there are exactly three right-angled triangles whose sides are integers while the area is numerically equal to twice the perimeter.
B2
1
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Putnam 1965 B2
In a round-robin tournament with
n
n
n
players
P
1
P_1
P
1
,
P
2
P_2
P
2
,
…
\ldots
…
,
P
n
P_n
P
n
(where
n
>
1
n > 1
n
>
1
), each player plays one game with each of the other players and the rules are such that no ties can occur. Let
w
r
w_r
w
r
and
l
r
l_r
l
r
be the number of games won and lost, respectively, by
P
r
P_r
P
r
. Show that
∑
r
=
1
n
w
r
2
=
∑
r
=
1
n
l
r
2
.
\sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2.
r
=
1
∑
n
w
r
2
=
r
=
1
∑
n
l
r
2
.
A6
1
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Putnam 1965 A6
In the plane with orthogonal Cartesian coordinates
x
x
x
and
y
y
y
, prove that the line whose equation is
u
x
+
v
y
=
1
ux+vy = 1
ux
+
v
y
=
1
will be tangent to the cirve
x
m
+
y
m
=
1
x^m+y^m=1
x
m
+
y
m
=
1
(where
m
>
1
m>1
m
>
1
) if and only if
u
n
+
v
n
=
1
u^n + v^n = 1
u
n
+
v
n
=
1
and
m
−
1
+
n
−
1
=
1
m^{-1} + n^{-1} = 1
m
−
1
+
n
−
1
=
1
.
A5
1
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Putnam 1965 A5
In how many ways can the integers from
1
1
1
to
n
n
n
be ordered subject to the condition that, except for the first integer on the left, every integer differs by
1
1
1
from some integer to the left of it?
A4
1
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Putnam 1965 A4
At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Prove that there are two couples
g
b
gb
g
b
and
g
′
b
′
g'b'
g
′
b
′
which dance whereas
b
b
b
does not dance with
g
′
g'
g
′
nor does
g
g
g
dance with
b
′
b'
b
′
.
A3
1
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Putnam 1965 A3
Show that, for any sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
of real numbers, the two conditions
lim
n
→
∞
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
)
n
=
α
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_n)}}n = \alpha
n
→
∞
lim
n
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
)
=
α
and
lim
n
→
∞
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
2
)
n
2
=
α
\lim_{n\to\infty}\frac{e^{(ia_1)} + e^{(ia_2)} + \cdots + e^{(ia_{n^2})}}{n^2} = \alpha
n
→
∞
lim
n
2
e
(
i
a
1
)
+
e
(
i
a
2
)
+
⋯
+
e
(
i
a
n
2
)
=
α
are equivalent.
A2
1
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Putnam 1965 A2
Show that, for any positive integer
n
n
n
,
∑
r
=
0
[
(
n
−
1
)
/
2
]
{
n
−
2
r
n
(
n
r
)
}
2
=
1
n
(
2
n
−
2
n
−
1
)
,
\sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1},
r
=
0
∑
[(
n
−
1
)
/2
]
{
n
n
−
2
r
(
r
n
)
}
2
=
n
1
(
n
−
1
2
n
−
2
)
,
where
[
x
]
[x]
[
x
]
means the greatest integer not exceeding
x
x
x
, and
(
n
r
)
\textstyle\binom nr
(
r
n
)
is the binomial coefficient "
n
n
n
choose
r
r
r
", with the convention
(
n
0
)
=
1
\textstyle\binom n0 = 1
(
0
n
)
=
1
.
B1
1
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Putnam 1965 B1
Evaluate \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.
A1
1
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Putnam 1965 A1
Let
A
B
C
ABC
A
BC
be a triangle with angle
A
<
A <
A
<
angle
C
<
9
0
∘
<
C < 90^\circ <
C
<
9
0
∘
<
angle
B
B
B
. Consider the bisectors of the external angles at
A
A
A
and
B
B
B
, each measured from the vertex to the opposoite side (extended). Suppose both of these line-segments are equal to
A
B
AB
A
B
. Compute the angle
A
A
A
.