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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1995 French Mathematical Olympiad
1995 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 4
1
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points in plane, AiBj=i+j
Suppose
A
1
,
A
2
,
A
3
,
B
1
,
B
2
,
B
3
A_1,A_2,A_3,B_1,B_2,B_3
A
1
,
A
2
,
A
3
,
B
1
,
B
2
,
B
3
are points in the plane such that for each
i
,
j
∈
{
1
,
2
,
3
}
i,j\in\{1,2,3\}
i
,
j
∈
{
1
,
2
,
3
}
it holds that
A
i
B
j
=
i
+
j
A_iB_j=i+j
A
i
B
j
=
i
+
j
. What can be said about these six points?
Problem 3
1
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three circles, minimal set of points
Consider three circles in the plane
Γ
1
,
Γ
2
,
Γ
3
\Gamma_1,\Gamma_2,\Gamma_3
Γ
1
,
Γ
2
,
Γ
3
of radii
R
R
R
passing through a point
O
O
O
, and denote by
D
\mathfrak D
D
the set of points of the plane which belong to at least two of these circles. Find the position of
Γ
1
,
Γ
2
,
Γ
3
\Gamma_1,\Gamma_2,\Gamma_3
Γ
1
,
Γ
2
,
Γ
3
for which the area of
D
\mathfrak D
D
is the minimum possible. Justify your answer.
Problem 2
1
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convergence of sequence
Study the convergence of a sequence defined by
u
0
≥
0
u_0\ge0
u
0
≥
0
and
u
n
+
1
=
u
n
+
1
n
+
1
u_{n+1}=\sqrt{u_n}+\frac1{n+1}
u
n
+
1
=
u
n
+
n
+
1
1
for all
n
∈
N
0
n\in\mathbb N_0
n
∈
N
0
.
Problem 5
1
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bijection over N
Let
f
f
f
be a bijection from
N
\mathbb N
N
to itself. Prove that one can always find three natural number
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
<
b
<
c
a<b<c
a
<
b
<
c
and
f
(
a
)
+
f
(
c
)
=
2
f
(
b
)
f(a)+f(c)=2f(b)
f
(
a
)
+
f
(
c
)
=
2
f
(
b
)
.
Problem 1
1
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centroid of triangle when line varies
We are given a triangle
A
B
C
ABC
A
BC
in a plane
P
P
P
. To any line
D
D
D
, not parallel to any side of the triangle, we associate the barycenter
G
D
G_D
G
D
of the set of intersection points of
D
D
D
with the sides of
△
A
B
C
\triangle ABC
△
A
BC
. The object of this problem is determining the set
F
\mathfrak F
F
of points
G
D
G_D
G
D
when
D
D
D
varies.(a) If
D
D
D
goes over all lines parallel to a given line
δ
\delta
δ
, prove that
G
D
G_D
G
D
describes a line
Δ
δ
\Delta_\delta
Δ
δ
. (b) Assume
△
A
B
C
\triangle ABC
△
A
BC
is equilateral. Prove that all lines
Δ
δ
\Delta_\delta
Δ
δ
are tangent to the same circle as
δ
\delta
δ
varies, and describe the set
F
\mathfrak F
F
. (c) If
A
B
C
ABC
A
BC
is an arbitrary triangle, prove that one can find a plane
P
P
P
and an equilateral triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
whose orthogonal projection onto
P
P
P
is
△
A
B
C
\triangle ABC
△
A
BC
, and describe the set
F
\mathfrak F
F
in the general case.