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Undergraduate contests
Putnam
1983 Putnam
1983 Putnam
Part of
Putnam
Subcontests
(12)
B5
1
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limit integral with distance to nearest integer
Let
∥
u
∥
\lVert u\rVert
∥
u
∥
denote the distance from the real number
u
u
u
to the nearest integer. For positive integers
n
n
n
, let
a
n
=
1
n
∫
1
n
∥
n
x
∥
d
x
.
a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.
a
n
=
n
1
∫
1
n
x
n
d
x
.
Determine
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
B3
1
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PDE (1+p(x))y''+y'q(x)+yr(x)=0 implies y'+Ayp(x)=Br(x)
Assume that the differential equation
y
′
′
′
+
p
(
x
)
y
′
′
+
q
(
x
)
y
′
+
r
(
x
)
y
=
0
y'''+p(x)y''+q(x)y'+r(x)y=0
y
′′′
+
p
(
x
)
y
′′
+
q
(
x
)
y
′
+
r
(
x
)
y
=
0
has solutions
y
1
(
x
)
y_1(x)
y
1
(
x
)
,
y
2
(
x
)
y_2(x)
y
2
(
x
)
,
y
3
(
x
)
y_3(x)
y
3
(
x
)
on the real line such that
y
1
(
x
)
2
+
y
2
(
x
)
2
+
y
3
(
x
)
2
=
1
y_1(x)^2+y_2(x)^2+y_3(x)^2=1
y
1
(
x
)
2
+
y
2
(
x
)
2
+
y
3
(
x
)
2
=
1
for all real
x
x
x
. Let
f
(
x
)
=
y
1
′
(
x
)
2
+
y
2
′
(
x
)
2
+
y
3
′
(
x
)
2
.
f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.
f
(
x
)
=
y
1
′
(
x
)
2
+
y
2
′
(
x
)
2
+
y
3
′
(
x
)
2
.
Find constants
A
A
A
and
B
B
B
such that
f
(
x
)
f(x)
f
(
x
)
is a solution to the differential equation
y
′
+
A
p
(
x
)
y
=
B
r
(
x
)
.
y'+Ap(x)y=Br(x).
y
′
+
A
p
(
x
)
y
=
B
r
(
x
)
.
B2
1
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representation as sum of powers of 2, as floor of polynomial
For positive integers
n
n
n
, let
C
(
n
)
C(n)
C
(
n
)
be the number of representation of
n
n
n
as a sum of nonincreasing powers of
2
2
2
, where no power can be used more than three times. For example,
C
(
8
)
=
5
C(8)=5
C
(
8
)
=
5
since the representations of
8
8
8
are:
8
,
4
+
4
,
4
+
2
+
2
,
4
+
2
+
1
+
1
,
and
2
+
2
+
2
+
1
+
1.
8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.
8
,
4
+
4
,
4
+
2
+
2
,
4
+
2
+
1
+
1
,
and
2
+
2
+
2
+
1
+
1.
Prove or disprove that there is a polynomial
P
(
x
)
P(x)
P
(
x
)
such that
C
(
n
)
=
⌊
P
(
n
)
⌋
C(n)=\lfloor P(n)\rfloor
C
(
n
)
=
⌊
P
(
n
)⌋
for all positive integers
n
n
n
.
B1
1
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region closer to one vertex than other vertices of cube
Let
v
v
v
be a vertex of a cube
C
C
C
with edges of length
4
4
4
. Let
S
S
S
be the largest sphere that can be inscribed in
C
C
C
. Let
R
R
R
be the region consisting of all points
p
p
p
between
S
S
S
and
C
C
C
such that
p
p
p
is closer to
v
v
v
than to any other vertex of the cube. Find the volume of
R
R
R
.
A6
1
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limit of double integral with exp
Let
F
(
x
)
=
x
4
exp
(
x
3
)
∫
0
x
∫
0
x
−
u
exp
(
u
3
+
v
3
)
d
v
d
u
.
F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.
F
(
x
)
=
exp
(
x
3
)
x
4
∫
0
x
∫
0
x
−
u
exp
(
u
3
+
v
3
)
d
v
d
u
.
Find
lim
x
→
∞
F
(
x
)
\lim_{x\to\infty}F(x)
lim
x
→
∞
F
(
x
)
or prove that it does not exist.
A5
1
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floor(u^n)-n is even for all n, does u exist?
Prove or disprove that there exists a positive real
u
u
u
such that
⌊
u
n
⌋
−
n
\lfloor u^n\rfloor-n
⌊
u
n
⌋
−
n
is an even integer for all positive integers
n
n
n
.
A4
1
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combinatorial sum is never zero
Let
k
k
k
be a positive integer and let
m
=
6
k
−
1
m=6k-1
m
=
6
k
−
1
. Let
S
(
m
)
=
∑
j
=
1
2
k
−
1
(
−
1
)
j
+
1
(
m
3
j
−
1
)
.
S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.
S
(
m
)
=
j
=
1
∑
2
k
−
1
(
−
1
)
j
+
1
(
3
j
−
1
m
)
.
Prove that
S
(
m
)
S(m)
S
(
m
)
is never zero.
A3
1
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1+2n+3n^2+...+(p-1)n^(p-2), F(a)!=F(b) mod p
Let
p
p
p
be an odd prime and let
F
(
n
)
=
1
+
2
n
+
3
n
2
+
…
+
(
p
−
1
)
n
p
−
2
.
F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.
F
(
n
)
=
1
+
2
n
+
3
n
2
+
…
+
(
p
−
1
)
n
p
−
2
.
Prove that if
a
a
a
and
b
b
b
are distinct integers in
{
0
,
1
,
2
,
…
,
p
−
1
}
\{0,1,2,\ldots,p-1\}
{
0
,
1
,
2
,
…
,
p
−
1
}
then
F
(
a
)
F(a)
F
(
a
)
and
F
(
b
)
F(b)
F
(
b
)
are not congruent modulo
p
p
p
.
A1
1
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exact divisor of at least one of 10^40, 20^30
How many positive integers
n
n
n
are there such that
n
n
n
is an exact divisors of at least one of the numbers
1
0
40
10^{40}
1
0
40
and
2
0
30
20^{30}
2
0
30
?
A2
1
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A fast clock
The shorthand of a clock has the length 3, the longhand has the length 4. Determine the distance between the endpoints of the hands at the time, where their distance increases the most.
B4
1
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Sequence contains a perfect square
Problem. Let
f
:
R
0
+
→
R
0
+
f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+
f
:
R
0
+
→
R
0
+
be a function defined as
f
(
n
)
=
n
+
⌊
n
⌋
∀
n
∈
R
0
+
.
f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.
f
(
n
)
=
n
+
⌊
n
⌋
∀
n
∈
R
0
+
.
Prove that for any positive integer
m
,
m,
m
,
the sequence
m
,
f
(
m
)
,
f
(
f
(
m
)
)
,
f
(
f
(
f
(
m
)
)
)
,
…
m,f(m),f(f(m)),f(f(f(m))),\ldots
m
,
f
(
m
)
,
f
(
f
(
m
))
,
f
(
f
(
f
(
m
)))
,
…
contains a perfect square.
B6
1
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Putnam 1983 B6
Let
k
k
k
be a positive integer, let m\equal{}2^k\plus{}1, and let
r
≠
1
r\neq 1
r
=
1
be a complex root of z^m\minus{}1\equal{}0. Prove that there exist polynomials
P
(
z
)
P(z)
P
(
z
)
and
Q
(
z
)
Q(z)
Q
(
z
)
with integer coefficients such that (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1.