MathDB
Problems
Contests
Undergraduate contests
IMC
2005 IMC
2005 IMC
Part of
IMC
Subcontests
(6)
6
2
Hide problems
IMC 2005 day 1 pb 6
6)
G
G
G
group,
G
m
G_{m}
G
m
and
G
n
G_{n}
G
n
commutative subgroups being the
m
m
m
and
n
n
n
th powers of the elements in
G
G
G
. Prove
G
g
c
d
(
m
,
n
)
G_{gcd(m,n)}
G
g
c
d
(
m
,
n
)
is commutative.
IMC 2005 day 2 pb 6
6. If
p
,
q
p,q
p
,
q
are rationals,
r
=
p
+
7
q
r=p+\sqrt{7}q
r
=
p
+
7
q
, then prove there exists a matrix
(
a
b
c
d
)
∈
M
2
(
Z
)
−
(
±
I
2
)
\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})
(
a
c
b
d
)
∈
M
2
(
Z
)
−
(
±
I
2
)
for which
a
r
+
b
c
r
+
d
=
r
\frac{ar+b}{cr+d}=r
cr
+
d
a
r
+
b
=
r
and
d
e
t
(
A
)
=
1
det(A)=1
d
e
t
(
A
)
=
1
5
2
Hide problems
IMC 2005 day1 pb 5
5) f twice cont diff,
∣
f
′
′
(
x
)
+
2
x
f
′
(
x
)
+
(
x
2
+
1
)
f
(
x
)
∣
≤
1
|f''(x)+2xf'(x)+(x^{2}+1)f(x)|\leq 1
∣
f
′′
(
x
)
+
2
x
f
′
(
x
)
+
(
x
2
+
1
)
f
(
x
)
∣
≤
1
. prove
lim
x
→
+
∞
f
(
x
)
=
0
\lim_{x\rightarrow +\infty} f(x) = 0
lim
x
→
+
∞
f
(
x
)
=
0
IMC 2005 day 2 pb 5
Find all
r
>
0
r > 0
r
>
0
such that when
f
:
R
2
→
R
f: \mathbb R^{2}\to \mathbb R
f
:
R
2
→
R
is differentiable, \|\textrm{grad} \; f(0,0)\| \equal{} 1, \|\textrm{grad} \; f(u) \minus{} \textrm{grad} \; f(v)\| \leq \| u \minus{} v\|, then the max of
f
f
f
on the disk
∥
u
∥
≤
r
\|u\|\leq r
∥
u
∥
≤
r
, is attained at exactly one point.
4
2
Hide problems
IMC 2005 day 1 pb 4
4) find all polynom with coeffs a permutation of
[
1
,
.
.
.
,
n
]
[1,...,n]
[
1
,
...
,
n
]
and all roots rational
IMC 2005 day 2 pb 4
Let
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
be a three times differentiable function. Prove that there exists
w
∈
[
−
1
,
1
]
w \in [-1,1]
w
∈
[
−
1
,
1
]
such that
f
′
′
′
(
w
)
6
=
f
(
1
)
2
−
f
(
−
1
)
2
−
f
′
(
0
)
.
\frac{f'''(w)}{6} = \frac{f(1)}{2}-\frac{f(-1)}{2}-f'(0).
6
f
′′′
(
w
)
=
2
f
(
1
)
−
2
f
(
−
1
)
−
f
′
(
0
)
.
3
2
Hide problems
IMC 2005 day 1 pb 3
3)
f
f
f
cont diff,
R
→
]
0
,
+
∞
[
R\rightarrow ]0,+\infty[
R
→
]
0
,
+
∞
[
, prove
∣
∫
0
1
f
3
−
f
(
0
)
2
∫
0
1
f
∣
≤
max
[
0
,
1
]
∣
f
′
∣
(
∫
0
1
f
)
2
|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}
∣
∫
0
1
f
3
−
f
(
0
)
2
∫
0
1
f
∣
≤
max
[
0
,
1
]
∣
f
′
∣
(
∫
0
1
f
)
2
IMC 2005 day 2 pb3
What is the maximal dimension of a linear subspace
V
V
V
of the vector space of real
n
×
n
n \times n
n
×
n
matrices such that for all
A
A
A
in
B
B
B
in
V
V
V
, we have \text{trace}\left(AB\right) \equal{} 0 ?
2
2
Hide problems
IMC 2005 day 1 pb 2
2) all elements in {0,1,2}; B[n] = number of rows with no 2 sequent 0's; A[n] with no 3 sequent elements the same; prove |A[n+1]|=3.|B[n]|
IMC 2005 day 2 pb 2
Let
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
be a function such that
(
f
(
x
)
)
n
(f(x))^{n}
(
f
(
x
)
)
n
is a polynomial for every integer
n
≥
2
n\geq 2
n
≥
2
. Is
f
f
f
also a polynomial?
1
2
Hide problems
IMC 2005 day 1 pb 1
Let
A
A
A
be a
n
×
n
n\times n
n
×
n
matrix such that
A
i
j
=
i
+
j
A_{ij} = i+j
A
ij
=
i
+
j
. Find the rank of
A
A
A
. [hide="Remark"]Not asked in the contest:
A
A
A
is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.
IMC 2005 day 2 pb1
1. Let
f
(
x
)
=
x
2
+
b
x
+
c
f(x)=x^2+bx+c
f
(
x
)
=
x
2
+
b
x
+
c
, M = {x | |f(x)|<1}. Prove
∣
M
∣
≤
2
2
|M|\leq 2\sqrt{2}
∣
M
∣
≤
2
2
(|...| = length of interval(s))