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Undergraduate contests
Putnam
1992 Putnam
1992 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Putnam 1992 B6
Let
M
M
M
be a set of real
n
×
n
n \times n
n
×
n
matrices such thati)
I
n
∈
M
I_{n} \in M
I
n
∈
M
, where
I
n
I_n
I
n
is the identity matrix.ii) If
A
∈
M
A\in M
A
∈
M
and
B
∈
M
B\in M
B
∈
M
, then either
A
B
∈
M
AB\in M
A
B
∈
M
or
−
A
B
∈
M
-AB\in M
−
A
B
∈
M
, but not bothiii) If
A
∈
M
A\in M
A
∈
M
and
B
∈
M
B \in M
B
∈
M
, then either
A
B
=
B
A
AB=BA
A
B
=
B
A
or
A
B
=
−
B
A
AB=-BA
A
B
=
−
B
A
.iv) If
A
∈
M
A\in M
A
∈
M
and
A
≠
I
n
A \ne I_n
A
=
I
n
, there is at least one
B
∈
M
B\in M
B
∈
M
such that
A
B
=
−
B
A
AB=-BA
A
B
=
−
B
A
.Prove that
M
M
M
contains at most
n
2
n^2
n
2
matrices.
B5
1
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Putnam 1992 B5
Let
D
n
D_n
D
n
denote the value of the
(
n
−
1
)
×
(
n
−
1
)
(n -1) \times (n - 1)
(
n
−
1
)
×
(
n
−
1
)
determinant
(
3
1
1
…
1
1
4
1
…
1
1
1
5
…
1
⋮
⋮
⋮
⋱
⋮
1
1
1
…
n
+
1
)
.
\begin{pmatrix} 3 & 1 &1 & \ldots & 1\\ 1 & 4 &1 & \ldots & 1\\ 1 & 1 & 5 & \ldots & 1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & 1 & \ldots & n+1 \end{pmatrix}.
3
1
1
⋮
1
1
4
1
⋮
1
1
1
5
⋮
1
…
…
…
⋱
…
1
1
1
⋮
n
+
1
.
Is the set
{
D
n
n
!
∣
n
≥
2
}
\left\{ \frac{D_n }{n!} \, | \, n \geq 2\right\}
{
n
!
D
n
∣
n
≥
2
}
bounded?
B4
1
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Putnam 1992 B4
Let
p
(
x
)
p(x)
p
(
x
)
be a nonzero polynomial of degree less than
1992
1992
1992
having no nonconstant factor in common with
x
3
−
x
x^3 -x
x
3
−
x
. Let
d
1992
d
x
1992
(
p
(
x
)
x
3
−
x
)
=
f
(
x
)
g
(
x
)
\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}
d
x
1992
d
1992
(
x
3
−
x
p
(
x
)
)
=
g
(
x
)
f
(
x
)
for polynomials
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
.
g(x).
g
(
x
)
.
Find the smallest possible degree of
f
(
x
)
f(x)
f
(
x
)
.
B3
1
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Putnam 1992 B3
For any pair
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers, a sequence
(
a
n
(
x
,
y
)
)
(a_{n}(x,y))
(
a
n
(
x
,
y
))
is defined as follows:
a
0
(
x
,
y
)
=
x
,
a
n
+
1
(
x
,
y
)
=
a
n
(
x
,
y
)
2
+
y
2
2
for
n
≥
0
a_{0}(x,y)=x, \;\;\;\; a_{n+1}(x,y) =\frac{a_{n}(x,y)^{2} +y^2 }{2} \;\, \text{for}\, n\geq 0
a
0
(
x
,
y
)
=
x
,
a
n
+
1
(
x
,
y
)
=
2
a
n
(
x
,
y
)
2
+
y
2
for
n
≥
0
Find the area of the region
{
(
x
,
y
)
∈
R
2
∣
(
a
n
(
x
,
y
)
)
converges
}
\{(x,y)\in \mathbb{R}^{2} \, |\, (a_{n}(x,y)) \,\, \text{converges} \}
{(
x
,
y
)
∈
R
2
∣
(
a
n
(
x
,
y
))
converges
}
.
B1
1
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Putnam 1992 B1
Let
S
S
S
be a set of
n
n
n
distinct real numbers. Let
A
S
A_{S}
A
S
be the set of numbers that occur as averages of two distinct elements of
S
S
S
. For a given
n
≥
2
n \geq 2
n
≥
2
, what is the smallest possible number of elements in
A
S
A_{S}
A
S
?
A5
1
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Putnam 1992 A5
For each positive integer
n
n
n
, let
a
n
=
0
a_n = 0
a
n
=
0
(or
1
1
1
) if the number of
1
1
1
’s in the binary representation of
n
n
n
is even (or odd), respectively. Show that there do not exist positive integers
k
k
k
and
m
m
m
such that
a
k
+
j
=
a
k
+
m
+
j
=
a
k
+
2
m
+
j
a_{k+j}=a_{k+m+j} =a_{k+2m+j}
a
k
+
j
=
a
k
+
m
+
j
=
a
k
+
2
m
+
j
for
0
≤
j
≤
m
−
1.
0 \leq j \leq m-1.
0
≤
j
≤
m
−
1.
A2
1
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Putnam 1992 A2
Define
C
(
α
)
C(\alpha)
C
(
α
)
to be the coefficient of
x
1992
x^{1992}
x
1992
in the power series about
x
=
0
x = 0
x
=
0
of
(
1
+
x
)
α
(1 + x)^{\alpha}
(
1
+
x
)
α
. Evaluate
∫
0
1
(
C
(
−
y
−
1
)
∑
k
=
1
1992
1
y
+
k
)
d
y
.
\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.
∫
0
1
(
C
(
−
y
−
1
)
k
=
1
∑
1992
y
+
k
1
)
d
y
.
A3
1
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Beautiful Problem
Let
m
,
n
m,n
m
,
n
are natural numbers such that
G
C
D
(
m
,
n
)
=
1
GCD(m,n)=1
GC
D
(
m
,
n
)
=
1
.Find all triplets
(
x
,
y
,
n
)
(x,y,n)
(
x
,
y
,
n
)
which sastify
(
x
2
+
y
2
)
m
=
(
x
y
)
n
(x^2+y^2)^m=(xy)^n
(
x
2
+
y
2
)
m
=
(
x
y
)
n
A1
1
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f ( f (n)) = f ( f (n+2)+2) = n
Find all functions
f
:
Z
→
Z
f : Z\rightarrow Z
f
:
Z
→
Z
for which we have f (0) \equal{} 1 and f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n, for every natural number
n
n
n
.
A4
1
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real-valued function
Let
f
f
f
be an infinitely differentiable real-valued function defined on the real numbers. If f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,..., Compute the values of the derivatives of f^{k}(0), k\equal{}0,1,2,3,...
A6
1
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Four points on the surface of a sphere
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?
B2
1
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Coefficient in (1+x+x^2+x^3)^n
For nonnegative integers
n
n
n
and
k
k
k
, define
Q
(
n
,
k
)
Q(n, k)
Q
(
n
,
k
)
to be the coefficient of
x
k
x^{k}
x
k
in the expansion
(
1
+
x
+
x
2
+
x
3
)
n
(1+x+x^{2}+x^{3})^{n}
(
1
+
x
+
x
2
+
x
3
)
n
. Prove that
Q
(
n
,
k
)
=
∑
j
=
0
k
(
n
j
)
(
n
k
−
2
j
)
Q(n, k) = \sum_{j=0}^{k}\binom{n}{j}\binom{n}{k-2j}
Q
(
n
,
k
)
=
∑
j
=
0
k
(
j
n
)
(
k
−
2
j
n
)
. [hide="hint"] Think of
(
n
j
)
\binom{n}{j}
(
j
n
)
as the number of ways you can pick the
x
2
x^{2}
x
2
term in the expansion.