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Undergraduate contests
Putnam
2000 Putnam
2000 Putnam
Part of
Putnam
Subcontests
(6)
6
2
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Putnam 2000 A6
Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with integer coefficients. Define a sequence
a
0
,
a
1
,
⋯
a_0, a_1, \cdots
a
0
,
a
1
,
⋯
of integers such that
a
0
=
0
a_0=0
a
0
=
0
and
a
n
+
1
=
f
(
a
n
)
a_{n+1}=f(a_n)
a
n
+
1
=
f
(
a
n
)
for all
n
≥
0
n \ge 0
n
≥
0
. Prove that if there exists a positive integer
m
m
m
for which
a
m
=
0
a_m=0
a
m
=
0
then either
a
1
=
0
a_1=0
a
1
=
0
or
a
2
=
0
a_2=0
a
2
=
0
.
Putnam 2000 B6
Let
B
B
B
be a set of more than
2
n
+
1
n
\tfrac{2^{n+1}}{n}
n
2
n
+
1
distinct points with coordinates of the form
(
±
1
,
±
1
,
⋯
,
±
1
)
(\pm 1, \pm 1, \cdots, \pm 1)
(
±
1
,
±
1
,
⋯
,
±
1
)
in
n
n
n
-dimensional space with
n
≥
3
n \ge 3
n
≥
3
. Show that there are three distinct points in
B
B
B
which are the vertices of an equilateral triangle.
5
2
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Putnam 2000 A5
Three distinct points with integer coordinates lie in the plane on a circle of radius
r
>
0
r>0
r
>
0
. Show that two of these points are separated by a distance of at least
r
1
/
3
r^{1/3}
r
1/3
.
Putnam 2000 B5
Let
S
0
S_0
S
0
be a finite set of positive integers. We define finite sets
S
1
,
S
2
,
⋯
S_1, S_2, \cdots
S
1
,
S
2
,
⋯
of positive integers as follows: the integer
a
a
a
in
S
n
+
1
S_{n+1}
S
n
+
1
if and only if exactly one of
a
−
1
a-1
a
−
1
or
a
a
a
is in
S
n
S_n
S
n
. Show that there exist infinitely many integers
N
N
N
for which
S
N
=
S
0
∪
{
N
+
a
:
a
∈
S
0
}
S_N = S_0 \cup \{ N + a: a \in S_0 \}
S
N
=
S
0
∪
{
N
+
a
:
a
∈
S
0
}
.
4
2
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Putnam 2000 A4
Show that the improper integral
lim
B
→
∞
∫
0
B
sin
(
x
)
sin
(
x
2
)
d
x
\lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx
B
→
∞
lim
∫
0
B
sin
(
x
)
sin
(
x
2
)
d
x
converges.
Putnam 2000 B4
Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function such that
f
(
2
x
2
−
1
)
=
2
x
f
(
x
)
f(2x^2-1)=2xf(x)
f
(
2
x
2
−
1
)
=
2
x
f
(
x
)
for all
x
x
x
. Show that
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for
−
1
≤
x
≤
1
-1\le x \le 1
−
1
≤
x
≤
1
.
3
2
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Putnam 2000 A3
The octagon
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P_1P_2P_3P_4P_5P_6P_7P_8
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon
P
1
P
3
P
5
P
7
P_1P_3P_5P_7
P
1
P
3
P
5
P
7
is a square of area
5
5
5
, and the polygon
P
2
P
4
P
6
P
8
P_2P_4P_6P_8
P
2
P
4
P
6
P
8
is a rectangle of area
4
4
4
, find the maximum possible area of the octagon.
Putnam 2000 B3
Let
f
(
t
)
=
∑
j
=
1
N
a
j
sin
(
2
π
j
t
)
f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)
f
(
t
)
=
j
=
1
∑
N
a
j
sin
(
2
πj
t
)
, where each
a
j
a_j
a
j
is areal and
a
N
a_N
a
N
is not equal to
0
0
0
. Let
N
k
N_k
N
k
denote the number of zeroes (including multiplicites) of
d
k
f
d
t
k
\dfrac{d^k f}{dt^k}
d
t
k
d
k
f
. Prove that
N
0
≤
N
1
≤
N
2
≤
⋯
and
lim
k
→
∞
N
k
=
2
N
.
N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N.
N
0
≤
N
1
≤
N
2
≤
⋯
and
k
→
∞
lim
N
k
=
2
N
.
[Only zeroes in [0, 1) should be counted.]
2
2
Hide problems
Putnam 2000 A2
Prove that there exist infinitely many integers
n
n
n
such that
n
n
n
,
n
+
1
n+1
n
+
1
,
n
+
2
n+2
n
+
2
are each the sum of the squares of two integers. [Example:
0
=
0
2
+
0
2
0=0^2+0^2
0
=
0
2
+
0
2
,
1
=
0
2
+
1
2
1=0^2+1^2
1
=
0
2
+
1
2
,
2
=
1
2
+
1
2
2=1^2+1^2
2
=
1
2
+
1
2
.]
Putnam 2000 B2
Prove that the expression
gcd
(
m
,
n
)
n
(
n
m
)
\dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m}
n
gcd
(
m
,
n
)
(
m
n
)
is an integer for all pairs of integers
n
≥
m
≥
1
n \ge m \ge 1
n
≥
m
≥
1
.
1
2
Hide problems
Putnam 2000 A1
Let
A
A
A
be a positive real number. What are the possible values of
∑
j
=
0
∞
x
j
2
,
\displaystyle\sum_{j=0}^{\infty} x_j^2,
j
=
0
∑
∞
x
j
2
,
given that
x
0
,
x
1
,
⋯
x_0, x_1, \cdots
x
0
,
x
1
,
⋯
are positive numbers for which
∑
j
=
0
∞
x
j
=
A
\displaystyle\sum_{j=0}^{\infty} x_j = A
j
=
0
∑
∞
x
j
=
A
?
Putnam 2000 B1
Let
a
j
a_j
a
j
,
b
j
b_j
b
j
,
c
j
c_j
c
j
be integers for
1
≤
j
≤
N
1 \le j \le N
1
≤
j
≤
N
. Assume for each
j
j
j
, at least one of
a
j
a_j
a
j
,
b
j
b_j
b
j
,
c
j
c_j
c
j
is odd. Show that there exists integers
r
,
s
,
t
r, s, t
r
,
s
,
t
such that
r
a
j
+
s
b
j
+
t
c
j
ra_j+sb_j+tc_j
r
a
j
+
s
b
j
+
t
c
j
is odd for at least
4
N
7
\tfrac{4N}{7}
7
4
N
values of
j
j
j
,
1
≤
j
≤
N
1 \le j \le N
1
≤
j
≤
N
.