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Problems
Contests
Undergraduate contests
IMC
2000 IMC
2000 IMC
Part of
IMC
Subcontests
(6)
6
2
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hard limit
Let
f
:
R
→
]
0
,
+
∞
[
f: \mathbb{R}\rightarrow ]0,+\infty[
f
:
R
→
]
0
,
+
∞
[
be an increasing differentiable function with
lim
x
→
+
∞
f
(
x
)
=
+
∞
\lim_{x\rightarrow+\infty}f(x)=+\infty
lim
x
→
+
∞
f
(
x
)
=
+
∞
and
f
′
f'
f
′
is bounded, and let
F
(
x
)
=
∫
0
x
f
(
t
)
d
t
F(x)=\int^x_0 f(t) dt
F
(
x
)
=
∫
0
x
f
(
t
)
d
t
. Define the sequence
(
a
n
)
(a_n)
(
a
n
)
recursively by
a
0
=
1
,
a
n
+
1
=
a
n
+
1
f
(
a
n
)
a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}
a
0
=
1
,
a
n
+
1
=
a
n
+
f
(
a
n
)
1
Define the sequence
(
b
n
)
(b_n)
(
b
n
)
by
b
n
=
F
−
1
(
n
)
b_n=F^{-1}(n)
b
n
=
F
−
1
(
n
)
. Prove that
lim
x
→
+
∞
(
a
n
−
b
n
)
=
0
\lim_{x\rightarrow+\infty}(a_n-b_n)=0
lim
x
→
+
∞
(
a
n
−
b
n
)
=
0
.
Nilpotent Matrices
Let
A
A
A
be a real
n
×
n
n\times n
n
×
n
Matrix and define
e
A
=
∑
k
=
0
∞
A
k
k
!
e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}
e
A
=
∑
k
=
0
∞
k
!
A
k
Prove or disprove that for any real polynomial
P
(
x
)
P(x)
P
(
x
)
and any real matrices
A
,
B
A,B
A
,
B
,
P
(
e
A
B
)
P(e^{AB})
P
(
e
A
B
)
is nilpotent if and only if
P
(
e
B
A
)
P(e^{BA})
P
(
e
B
A
)
is nilpotent.
5
2
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ring of characteristic zero (remarkably easy for a q5)
Let
R
R
R
be a ring of characteristic zero. Let
e
,
f
,
g
∈
R
e,f,g\in R
e
,
f
,
g
∈
R
be idempotent elements (an element
x
x
x
is called idempotent if
x
2
=
x
x^2=x
x
2
=
x
) satisfying
e
+
f
+
g
=
0
e+f+g=0
e
+
f
+
g
=
0
. Show that
e
=
f
=
g
=
0
e=f=g=0
e
=
f
=
g
=
0
.
functional equation
Find all functions
R
+
→
R
+
\mathbb{R}^+\rightarrow\mathbb{R}^+
R
+
→
R
+
for which we have for all
x
,
y
∈
R
+
x,y\in \mathbb{R}^+
x
,
y
∈
R
+
that
f
(
x
)
f
(
y
f
(
x
)
)
=
f
(
x
+
y
)
f(x)f(yf(x))=f(x+y)
f
(
x
)
f
(
y
f
(
x
))
=
f
(
x
+
y
)
.
4
1
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Inequality on sum
Let
(
x
i
)
(x_i)
(
x
i
)
be a decreasing sequence of positive reals, then show that: (a) for every positive integer
n
n
n
we have
∑
i
=
1
n
x
i
2
≤
∑
i
=
1
n
x
i
i
\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}
∑
i
=
1
n
x
i
2
≤
∑
i
=
1
n
i
x
i
. (b) there is a constant C for which we have
∑
k
=
1
∞
1
k
∑
i
=
k
∞
x
i
2
≤
C
∑
i
=
1
∞
x
i
\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i
∑
k
=
1
∞
k
1
∑
i
=
k
∞
x
i
2
≤
C
∑
i
=
1
∞
x
i
.
2
2
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find all pairs
Let
p
(
x
)
=
x
5
+
x
p(x)=x^5+x
p
(
x
)
=
x
5
+
x
and
q
(
x
)
=
x
5
+
x
2
q(x)=x^5+x^2
q
(
x
)
=
x
5
+
x
2
, Find al pairs
(
w
,
z
)
∈
C
×
C
(w,z)\in \mathbb{C}\times\mathbb{C}
(
w
,
z
)
∈
C
×
C
,
w
≠
z
w\not=z
w
=
z
for which
p
(
w
)
=
p
(
z
)
,
q
(
w
)
=
q
(
z
)
p(w)=p(z),q(w)=q(z)
p
(
w
)
=
p
(
z
)
,
q
(
w
)
=
q
(
z
)
.
continuous but nowhere monotone function
Let
f
f
f
be continuous and nowhere monotone on
[
0
,
1
]
[0,1]
[
0
,
1
]
. Show that the set of points on which
f
f
f
obtains a local minimum is dense.
3
2
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Complex matrices
Let
A
,
B
∈
C
n
×
n
A,B\in\mathbb{C}^{n\times n}
A
,
B
∈
C
n
×
n
with
ρ
(
A
B
−
B
A
)
=
1
\rho(AB - BA) = 1
ρ
(
A
B
−
B
A
)
=
1
. Show that
(
A
B
−
B
A
)
2
=
0
(AB - BA)^2 = 0
(
A
B
−
B
A
)
2
=
0
.
polynomial with complex coefficients
Let
p
(
z
)
p(z)
p
(
z
)
be a polynomial of degree
n
>
0
n>0
n
>
0
with complex coefficients. Prove that there are at least
n
+
1
n+1
n
+
1
complex numbers
z
z
z
for which
p
(
z
)
∈
{
0
,
1
}
p(z)\in \{0,1\}
p
(
z
)
∈
{
0
,
1
}
.
1
2
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A fixed point
Does every monotone increasing function
f
:
[
0
,
1
]
→
[
0
,
1
]
f : \mathbb[0,1] \rightarrow \mathbb[0,1]
f
:
[
0
,
1
]
→
[
0
,
1
]
have a fixed point? What about every monotone decreasing function?
Dividing a square
Show that a square may be partitioned into
n
n
n
smaller squares for sufficiently large
n
n
n
. Show that for some constant
N
(
d
)
N(d)
N
(
d
)
, a
d
d
d
-dimensional cube can be partitioned into
n
n
n
smaller cubes if
n
≥
N
(
d
)
n \geq N(d)
n
≥
N
(
d
)
.