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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1991 French Mathematical Olympiad
1991 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
roots of unity, geometrical and polynomial
(a) For given complex numbers
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
, we define a function
P
:
C
→
C
P:\mathbb C\to\mathbb C
P
:
C
→
C
by
P
(
z
)
=
z
5
+
a
4
z
4
+
a
3
z
3
+
a
2
z
2
+
a
1
z
P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z
P
(
z
)
=
z
5
+
a
4
z
4
+
a
3
z
3
+
a
2
z
2
+
a
1
z
. Let
w
k
=
e
2
k
i
π
/
5
w_k=e^{2ki\pi/5}
w
k
=
e
2
kiπ
/5
, where
k
=
0
,
…
,
4
k=0,\ldots,4
k
=
0
,
…
,
4
. Prove that
P
(
w
0
)
+
P
(
w
1
)
+
P
(
w
2
)
+
P
(
w
3
)
+
P
(
w
4
)
=
5.
P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.
P
(
w
0
)
+
P
(
w
1
)
+
P
(
w
2
)
+
P
(
w
3
)
+
P
(
w
4
)
=
5.
(b) Let
A
1
,
A
2
,
A
3
,
A
4
,
A
5
A_1,A_2,A_3,A_4,A_5
A
1
,
A
2
,
A
3
,
A
4
,
A
5
be five points in the plane. A pentagon is inscribed in the circle with center
A
1
A_1
A
1
and radius
R
R
R
. Prove that there is a vertex
S
S
S
of the pentagon for which
S
A
1
⋅
S
A
2
⋅
S
A
3
⋅
S
A
4
⋅
S
A
5
≥
R
5
.
SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.
S
A
1
⋅
S
A
2
⋅
S
A
3
⋅
S
A
4
⋅
S
A
5
≥
R
5
.
Problem 4
1
Hide problems
if x∈A then 2x∉A, maximize |A|
Let
p
p
p
be a nonnegative integer and let
n
=
2
p
n=2^p
n
=
2
p
. Consider all subsets
A
A
A
of the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
with the property that, whenever
x
∈
A
x\in A
x
∈
A
,
2
x
∉
A
2x\notin A
2
x
∈
/
A
. Find the maximum number of elements that such a set
A
A
A
can have.
Problem 3
1
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inscribed tetrahedra with orthogonal segments
Let
S
S
S
be a fixed point on a sphere
Σ
\Sigma
Σ
with center
Ω
\Omega
Ω
. Consider all tetrahedra
S
A
B
C
SABC
S
A
BC
inscribed in
Σ
\Sigma
Σ
such that
S
A
,
S
B
,
S
C
SA,SB,SC
S
A
,
SB
,
SC
are pairwise orthogonal. (a) Prove that all the planes
A
B
C
ABC
A
BC
pass through a single point. (b) In one such tetrahedron,
H
H
H
and
O
O
O
are the orthogonal projections of
S
S
S
and
Ω
\Omega
Ω
onto the plane
A
B
C
ABC
A
BC
, respectively. Let
R
R
R
denote the circumradius of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that
R
2
=
O
H
2
+
2
S
H
2
R^2=OH^2+2SH^2
R
2
=
O
H
2
+
2
S
H
2
.
Problem 2
1
Hide problems
limits of function on two variables
For each
n
∈
N
n\in\mathbb N
n
∈
N
, the function
f
n
f_n
f
n
is defined on real numbers
x
≥
n
x\ge n
x
≥
n
by
f
n
(
x
)
=
x
−
n
+
x
−
n
+
1
+
…
+
x
+
n
−
(
2
n
+
1
)
x
.
f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.
f
n
(
x
)
=
x
−
n
+
x
−
n
+
1
+
…
+
x
+
n
−
(
2
n
+
1
)
x
.
(a) If
n
n
n
is fixed, prove that
lim
x
→
+
∞
f
n
(
x
)
=
0
\lim_{x\to+\infty}f_n(x)=0
lim
x
→
+
∞
f
n
(
x
)
=
0
. (b) Find the limit of
f
n
(
n
)
f_n(n)
f
n
(
n
)
as
n
→
+
∞
n\to+\infty
n
→
+
∞
.
Problem 1
1
Hide problems
1^p+2^p+...+n^p square
(a) Suppose that
x
n
(
n
≥
0
)
x_n~(n\ge0)
x
n
(
n
≥
0
)
is a sequence of real numbers with the property that
x
0
3
+
x
1
3
+
…
+
x
n
3
=
(
x
0
+
x
1
+
…
+
x
n
)
2
x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2
x
0
3
+
x
1
3
+
…
+
x
n
3
=
(
x
0
+
x
1
+
…
+
x
n
)
2
for each
n
∈
N
n\in\mathbb N
n
∈
N
. Prove that for each
n
∈
N
0
n\in\mathbb N_0
n
∈
N
0
there exists
m
∈
N
0
m\in\mathbb N_0
m
∈
N
0
such that
x
0
+
x
1
+
…
+
x
n
=
m
(
m
+
1
)
2
x_0+x_1+\ldots+x_n=\frac{m(m+1)}2
x
0
+
x
1
+
…
+
x
n
=
2
m
(
m
+
1
)
. (b) For natural numbers
n
n
n
and
p
p
p
, we define
S
n
,
p
=
1
p
+
2
p
+
…
+
n
p
S_{n,p}=1^p+2^p+\ldots+n^p
S
n
,
p
=
1
p
+
2
p
+
…
+
n
p
. Find all natural numbers
p
p
p
such that
S
n
,
p
S_{n,p}
S
n
,
p
is a perfect square for each
n
∈
N
n\in\mathbb N
n
∈
N
.