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Problems
Contests
Undergraduate contests
Putnam
1990 Putnam
1990 Putnam
Part of
Putnam
Subcontests
(12)
B6
1
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Lines on a Plane
Let
S
S
S
be a nonempty closed bounded convex set in the plane. Let
K
K
K
be a line and
t
t
t
a positive number. Let
L
1
L_1
L
1
and
L
2
L_2
L
2
be support lines for
S
S
S
parallel to
K
1
K_1
K
1
, and let
L
‾
\overline {L}
L
be the line parallel to
K
K
K
and midway between
L
1
L_1
L
1
and
L
2
L_2
L
2
. Let
B
S
(
K
,
t
)
B_S(K,t)
B
S
(
K
,
t
)
be the band of points whose distance from
L
‾
\overline{L}
L
is at most
(
t
2
)
w
\left( \frac {t}{2} \right) w
(
2
t
)
w
, where
w
w
w
is the distance between
L
1
L_1
L
1
and
L
2
L_2
L
2
. What is the smallest
t
t
t
such that
S
∩
⋂
K
B
S
(
K
,
t
)
≠
∅
S \cap \bigcap_K B_S (K, t) \ne \emptyset
S
∩
K
⋂
B
S
(
K
,
t
)
=
∅
for all
S
S
S
? (
K
K
K
runs over all lines in the plane.)
B5
1
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alpha_i such that polynomial has all real/distinct roots
Is there an infinite sequence
a
0
,
a
1
,
a
2
,
⋯
a_0, a_1, a_2, \cdots
a
0
,
a
1
,
a
2
,
⋯
of nonzero real numbers such that for
n
=
1
,
2
,
3
,
⋯
n = 1, 2, 3, \cdots
n
=
1
,
2
,
3
,
⋯
the polynomial
p
n
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n
p
n
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
has exactly
n
n
n
distinct real roots?
B4
1
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g and h: Group generators
Let
G
G
G
be a finite group of order
n
n
n
generated by
a
a
a
and
b
b
b
. Prove or disprove: there is a sequence
g
1
,
g
2
,
g
3
,
⋯
,
g
2
n
g_1, g_2, g_3, \cdots, g_{2n}
g
1
,
g
2
,
g
3
,
⋯
,
g
2
n
such that:
(
1
)
(1)
(
1
)
every element of
G
G
G
occurs exactly twice, and
(
2
)
(2)
(
2
)
g
i
+
1
g_{i+1}
g
i
+
1
equals
g
i
a
g_{i}a
g
i
a
or
g
i
b
g_ib
g
i
b
for
i
=
1
,
2
,
⋯
,
2
n
i = 1, 2, \cdots, 2n
i
=
1
,
2
,
⋯
,
2
n
. (interpret
g
2
n
+
1
g_{2n+1}
g
2
n
+
1
as
g
1
g_1
g
1
.)
B3
1
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A set of 2-by-2 matrices
Let
S
S
S
be a set of
2
×
2
2 \times 2
2
×
2
integer matrices whose entries
a
i
j
(
1
)
a_{ij}(1)
a
ij
(
1
)
are all squares of integers and,
(
2
)
(2)
(
2
)
satisfy
a
i
j
≤
200
a_{ij} \le 200
a
ij
≤
200
. Show that
S
S
S
has more than
50387
(
=
1
5
4
−
1
5
2
−
15
+
2
)
50387 (=15^4-15^2-15+2)
50387
(
=
1
5
4
−
1
5
2
−
15
+
2
)
elements, then it has two elements that commute.
B2
1
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P_n(n,x): given a product, prove a sum
Prove that for
∣
x
∣
<
1
|x| < 1
∣
x
∣
<
1
,
∣
z
∣
>
1
|z| > 1
∣
z
∣
>
1
,
1
+
∑
j
=
1
∞
(
1
+
x
j
)
P
j
=
0
,
1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0,
1
+
j
=
1
∑
∞
(
1
+
x
j
)
P
j
=
0
,
where
P
j
P_j
P
j
is
(
1
−
z
)
(
1
−
z
x
)
(
1
−
z
x
2
)
⋯
(
1
−
z
x
j
−
1
)
(
z
−
x
)
(
z
−
x
2
)
(
z
−
x
3
)
⋯
(
z
−
x
j
)
.
\dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}.
(
z
−
x
)
(
z
−
x
2
)
(
z
−
x
3
)
⋯
(
z
−
x
j
)
(
1
−
z
)
(
1
−
z
x
)
(
1
−
z
x
2
)
⋯
(
1
−
z
x
j
−
1
)
.
B1
1
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Integral and Derivative Equation
Find all real-valued continuously differentiable functions
f
f
f
on the real line such that for all
x
x
x
,
(
f
(
x
)
)
2
=
∫
0
x
[
(
f
(
t
)
)
2
+
(
f
′
(
t
)
)
2
]
d
t
+
1990.
\left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990.
(
f
(
x
)
)
2
=
∫
0
x
[
(
f
(
t
)
)
2
+
(
f
′
(
t
)
)
2
]
d
t
+
1990.
A6
1
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Subset Ordered Pairs of {1, 2, ..., 10}
If
X
X
X
is a finite set, let
X
X
X
denote the number of elements in
X
X
X
. Call an ordered pair
(
S
,
T
)
(S,T)
(
S
,
T
)
of subsets of
{
1
,
2
,
⋯
,
n
}
\{ 1, 2, \cdots, n \}
{
1
,
2
,
⋯
,
n
}
\emph {admissible} if
s
>
∣
T
∣
s > |T|
s
>
∣
T
∣
for each
s
∈
S
s \in S
s
∈
S
, and
t
>
∣
S
∣
t > |S|
t
>
∣
S
∣
for each
t
∈
T
t \in T
t
∈
T
. How many admissible ordered pairs of subsets
{
1
,
2
,
⋯
,
10
}
\{ 1, 2, \cdots, 10 \}
{
1
,
2
,
⋯
,
10
}
are there? Prove your answer.
A5
1
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Square matrices A and B
If
A
\mathbf{A}
A
and
B
\mathbf{B}
B
are square matrices of the same size such that
A
B
A
B
=
0
\mathbf{ABAB}=\mathbf{0}
ABAB
=
0
, does it follow that
B
A
B
A
=
0
\mathbf{BABA}=\mathbf{0}
BABA
=
0
.
A3
1
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Convex Lattice Pentagon of Area at least 5/2
Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to
5
2
\dfrac {5}{2}
2
5
.
A2
1
Hide problems
Subsequence that converges to sqrt(2)
Is
2
\sqrt{2}
2
the limit of a sequence of numbers of the form
n
3
−
m
3
\sqrt[3]{n} - \sqrt[3]{m}
3
n
−
3
m
, where
n
,
m
=
0
,
1
,
2
,
⋯
n, m = 0, 1, 2, \cdots
n
,
m
=
0
,
1
,
2
,
⋯
.
A1
1
Hide problems
Sum of well-known sequences
Let
T
0
=
2
,
T
1
=
3
,
T
2
=
6
,
T_0=2, T_1=3, T_2=6,
T
0
=
2
,
T
1
=
3
,
T
2
=
6
,
and for
n
≥
3
n\ge 3
n
≥
3
,
T
n
=
(
n
+
4
)
T
n
−
1
−
4
n
T
n
−
2
+
(
4
n
−
8
)
T
n
−
3
.
T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.
T
n
=
(
n
+
4
)
T
n
−
1
−
4
n
T
n
−
2
+
(
4
n
−
8
)
T
n
−
3
.
The first few terms are
2
,
3
,
6
,
14
,
40
,
152
,
784
,
5158
,
40576
,
363392.
2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.
2
,
3
,
6
,
14
,
40
,
152
,
784
,
5158
,
40576
,
363392.
Find a formula for
T
n
T_n
T
n
of the form
T
n
=
A
n
+
B
n
,
T_n=A_n+B_n,
T
n
=
A
n
+
B
n
,
where
{
A
n
}
\{A_n\}
{
A
n
}
and
{
B
n
}
\{B_n\}
{
B
n
}
are well known sequences.
A4
1
Hide problems
putnam 1990
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point?