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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1996 French Mathematical Olympiad
1996 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
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existence of sequence of positive integers with pairwise sums
Let
n
n
n
be a positive integer. We say that a natural number
k
k
k
has the property
C
n
C_n
C
n
if there exist
2
k
2k
2
k
distinct positive integers
a
1
,
b
1
,
…
,
a
k
,
b
k
a_1,b_1,\ldots,a_k,b_k
a
1
,
b
1
,
…
,
a
k
,
b
k
such that the sums
a
1
+
b
1
,
…
,
a
k
+
b
k
a_1+b_1,\ldots,a_k+b_k
a
1
+
b
1
,
…
,
a
k
+
b
k
are distinct and strictly smaller than
n
n
n
.(a) Prove that if
k
k
k
has the property
C
n
C_n
C
n
then
k
≤
2
n
−
3
5
k\le \frac{2n-3}{5}
k
≤
5
2
n
−
3
. (b) Prove that
5
5
5
has the property
C
14
C_{14}
C
14
. (c) If
(
2
n
−
3
)
/
5
(2n-3)/5
(
2
n
−
3
)
/5
is an integer, prove that it has the property
C
n
C_n
C
n
.
Problem 4
1
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x^y+y^x>1, x^x minimum
(a) A function
f
f
f
is defined by
f
(
x
)
=
x
x
f(x)=x^x
f
(
x
)
=
x
x
for all
x
>
0
x>0
x
>
0
. Find the minimum value of
f
f
f
. (b) If
x
x
x
and
y
y
y
are two positive real numbers, show that
x
y
+
y
x
>
1
x^y+y^x>1
x
y
+
y
x
>
1
.
Problem 3
1
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rectangular parallelepiped
(a) Let there be given a rectangular parallelepiped. Show that some four of its vertices determine a tetrahedron whose all faces are right triangles. (b) Conversely, prove that every tetrahedron whose all faces are right triangles can be obtained by selecting four vertices of a rectangular parallelepiped. (c) Now investigate such tetrahedra which also have at least two isosceles faces. Given the length
a
a
a
of the shortest edge, compute the lengths of the other edges.
Problem 2
1
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sequence, prove periodicity and existence of sufficiently small element
Let
a
a
a
be an odd natural number and
b
b
b
be a positive integer. We define a sequence of reals
(
u
n
)
(u_n)
(
u
n
)
as follows:
u
0
=
b
u_0=b
u
0
=
b
and, for all
n
∈
N
0
n\in\mathbb N_0
n
∈
N
0
,
u
n
+
1
u_{n+1}
u
n
+
1
is
u
n
2
\frac{u_n}2
2
u
n
if
u
n
u_n
u
n
is even and
a
+
u
n
a+u_n
a
+
u
n
otherwise.(a) Prove that one can find an element of
u
n
u_n
u
n
smaller than
a
a
a
. (b) Prove that the sequence is eventually periodic.
Problem 1
1
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triangle and points making squares
Consider a triangle
A
B
C
ABC
A
BC
and points
D
,
E
,
F
,
G
,
H
,
I
D,E,F,G,H,I
D
,
E
,
F
,
G
,
H
,
I
in the plane such that
A
B
E
D
ABED
A
BE
D
,
B
C
G
F
BCGF
BCGF
and
A
C
H
I
ACHI
A
C
H
I
are squares exterior to the triangle. Prove that points
D
,
E
,
F
,
G
,
H
,
I
D,E,F,G,H,I
D
,
E
,
F
,
G
,
H
,
I
are concyclic if and only if one of the following two statements hold:(i)
A
B
C
ABC
A
BC
is an equilateral triangle. (ii)
A
B
C
ABC
A
BC
is an isosceles right triangle.