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Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1992 French Mathematical Olympiad
1992 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
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floor((10^1992)/(10^83+7)), number of digits 1
Determine the number of digits
1
1
1
in the integer part of
1
0
1992
1
0
83
+
7
\frac{10^{1992}}{10^{83}+7}
1
0
83
+
7
1
0
1992
.
Problem 4
1
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sequence, convergent and monotonous
Given
u
0
,
u
1
u_0,u_1
u
0
,
u
1
with
0
<
u
0
,
u
1
<
1
0<u_0,u_1<1
0
<
u
0
,
u
1
<
1
, define the sequence
(
u
n
)
(u_n)
(
u
n
)
recurrently by the formula
u
n
+
2
=
1
2
(
u
n
+
1
+
u
n
)
.
u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).
u
n
+
2
=
2
1
(
u
n
+
1
+
u
n
)
.
(a) Prove that the sequence
u
n
u_n
u
n
is convergent and find its limit. (b) Prove that, starting from some index
n
0
n_0
n
0
, the sequence
u
n
u_n
u
n
is monotonous.
Problem 3
1
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tetrahedron, barycenter / incenter / circumcenter statements equivalent
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron inscribed in a sphere with center
O
O
O
, and
G
G
G
and
I
I
I
be its barycenter and incenter respectively. Prove that the following are equivalent: (i) Points
O
O
O
and
G
G
G
coincide. (ii) The four faces of the tetrahedron are congruent. (iii) Points
O
O
O
and
I
I
I
coincide.
Problem 2
1
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inscribed 3/4-gon, maximize sum of squares of sides
Let
C
\mathcal C
C
be a circle of radius
1
1
1
. (a) Determine the triangles
A
B
C
ABC
A
BC
inscribed in
C
\mathcal C
C
for which
A
B
2
+
B
C
2
+
C
A
2
AB^2+BC^2+CA^2
A
B
2
+
B
C
2
+
C
A
2
is maximal. (b) Determine the quadrilaterals
A
B
C
D
ABCD
A
BC
D
inscribed in
C
\mathcal C
C
for which
A
B
2
+
A
C
2
+
A
D
2
+
B
C
2
+
B
D
2
+
C
D
2
AB^2+AC^2+AD^2+BC^2+BD^2+CD^2
A
B
2
+
A
C
2
+
A
D
2
+
B
C
2
+
B
D
2
+
C
D
2
is maximal.
Problem 1
1
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properties of set of points in plane, vector equality
Let
Δ
\Delta
Δ
be a convex figure in a plane
P
\mathcal P
P
. Given a point
A
∈
P
A\in\mathcal P
A
∈
P
, to each pair
(
M
,
N
)
(M,N)
(
M
,
N
)
of points in
Δ
\Delta
Δ
we associate the point
m
∈
P
m\in\mathcal P
m
∈
P
such that
A
m
→
=
M
N
→
2
\overrightarrow{Am}=\frac{\overrightarrow{MN}}2
A
m
=
2
MN
and denote by
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
the set of all so obtained points
m
m
m
.(a) i. Prove that
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
is centrally symmetric. ii. Under which conditions is
δ
A
(
Δ
)
=
Δ
\delta_A(\Delta)=\Delta
δ
A
(
Δ
)
=
Δ
? iii. Let
B
,
C
B,C
B
,
C
be points in
P
\mathcal P
P
. Find a transformation which sends
δ
B
(
Δ
)
\delta_B(\Delta)
δ
B
(
Δ
)
to
δ
C
(
Δ
)
\delta_C(\Delta)
δ
C
(
Δ
)
. (b) Determine
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
if i.
Δ
\Delta
Δ
is a set in the plane determined by two parallel lines. ii.
Δ
\Delta
Δ
is bounded by a triangle. iii.
Δ
\Delta
Δ
is a semi-disk. (c) Prove that in the cases
b
.
2
b.2
b
.2
and
b
.
3
b.3
b
.3
the lengths of the boundaries of
Δ
\Delta
Δ
and
δ
A
(
Δ
)
\delta_A(\Delta)
δ
A
(
Δ
)
are equal.