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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1993 French Mathematical Olympiad
1993 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
maxmizing volume in tetrahedron
(a) Let there be two given points
A
,
B
A,B
A
,
B
in the plane. i. Find the triangles
M
A
B
MAB
M
A
B
with the given area and the minimal perimeter. ii. Find the triangles
M
A
B
MAB
M
A
B
with a given perimeter and the maximal area. (b) In a tetrahedron of volume
V
V
V
, let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be the lengths of its four edges, no three of which are coplanar, and let
L
=
a
+
b
+
c
+
d
L=a+b+c+d
L
=
a
+
b
+
c
+
d
. Determine the maximum value of
V
L
3
\frac V{L^3}
L
3
V
.
Problem 4
1
Hide problems
covering a disk with other disks
We are given a disk
D
\mathcal D
D
of radius
1
1
1
in the plane.(a) Prove that
D
\mathcal D
D
cannot be covered with two disks of radii
r
<
1
r<1
r
<
1
. (b) Prove that, for some
r
<
1
r<1
r
<
1
,
D
\mathcal D
D
can be covered with three disks of radius
r
r
r
. What is the smallest such
r
r
r
?
Problem 3
1
Hide problems
f(n)≤(f(n-1)+f(n+1))/2 from Z->R, f constant
Let
f
f
f
be a function from
Z
\mathbb Z
Z
to
R
\mathbb R
R
which is bounded from above and satisfies
f
(
n
)
≤
1
2
(
f
(
n
−
1
)
+
f
(
n
+
1
)
)
f(n)\le\frac12(f(n-1)+f(n+1))
f
(
n
)
≤
2
1
(
f
(
n
−
1
)
+
f
(
n
+
1
))
for all
n
n
n
. Show that
f
f
f
is constant.
Problem 2
1
Hide problems
existence of increasing sequences
Let
n
n
n
be a given positive integer.(a) Do there exist
2
n
+
1
2n+1
2
n
+
1
consecutive positive integers
a
0
,
a
1
,
…
,
a
2
n
a_0,a_1,\ldots,a_{2n}
a
0
,
a
1
,
…
,
a
2
n
in the ascending order such that
a
0
+
a
1
+
…
+
a
n
=
a
n
+
1
+
…
+
a
2
n
a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}
a
0
+
a
1
+
…
+
a
n
=
a
n
+
1
+
…
+
a
2
n
? (b) Do there exist consecutive positive integers
a
0
,
a
+
1
,
…
,
a
2
n
a_0,a+1,\ldots,a_{2n}
a
0
,
a
+
1
,
…
,
a
2
n
in ascending order such that
a
0
2
+
a
1
2
+
…
+
a
n
2
=
a
n
+
1
2
+
…
+
a
2
n
2
a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2
a
0
2
+
a
1
2
+
…
+
a
n
2
=
a
n
+
1
2
+
…
+
a
2
n
2
? (c) Do there exist consecutive positive integers
a
0
,
a
1
,
…
,
a
2
n
a_0,a_1,\ldots,a_{2n}
a
0
,
a
1
,
…
,
a
2
n
in ascending order such that
a
0
3
+
a
1
3
+
…
+
a
n
3
=
a
n
+
1
3
+
…
+
a
2
n
3
a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3
a
0
3
+
a
1
3
+
…
+
a
n
3
=
a
n
+
1
3
+
…
+
a
2
n
3
?[hide=Official Hint]You may study the function
f
(
x
)
=
(
x
−
n
)
3
+
…
+
x
3
−
(
x
+
1
)
3
−
…
−
(
x
+
n
)
3
f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3
f
(
x
)
=
(
x
−
n
)
3
+
…
+
x
3
−
(
x
+
1
)
3
−
…
−
(
x
+
n
)
3
and prove that the equation
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
has a unique solution
x
n
x_n
x
n
with
3
n
(
n
+
1
)
<
x
n
<
3
n
(
n
+
1
)
+
1
3n(n+1)<x_n<3n(n+1)+1
3
n
(
n
+
1
)
<
x
n
<
3
n
(
n
+
1
)
+
1
. You may use the identity
1
3
+
2
3
+
…
+
n
3
=
n
2
(
n
+
1
)
2
2
1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2
1
3
+
2
3
+
…
+
n
3
=
2
n
2
(
n
+
1
)
2
.
Problem 1
1
Hide problems
measuring with set of weights
Assume we are given a set of weights,
x
1
x_1
x
1
of which have mass
d
1
d_1
d
1
,
x
2
x_2
x
2
have mass
d
2
d_2
d
2
, etc,
x
k
x_k
x
k
have mass
d
k
d_k
d
k
, where
x
i
,
d
i
x_i,d_i
x
i
,
d
i
are positive integers and
1
≤
d
1
<
d
2
<
…
<
d
k
1\le d_1<d_2<\ldots<d_k
1
≤
d
1
<
d
2
<
…
<
d
k
. Let us denote their total sum by
n
=
x
1
d
1
+
…
+
x
k
d
k
n=x_1d_1+\ldots+x_kd_k
n
=
x
1
d
1
+
…
+
x
k
d
k
. We call such a set of weights perfect if each mass
0
,
1
,
…
,
n
0,1,\ldots,n
0
,
1
,
…
,
n
can be uniquely obtained using these weights.(a) Write down all sets of weights of total mass
5
5
5
. Which of them are perfect? (b) Show that a perfect set of weights satisfies
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
k
)
=
n
+
1.
(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
k
)
=
n
+
1.
(c) Conversely, if
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
k
)
=
n
+
1
(1+x_1)(1+x_2)\cdots(1+x_k)=n+1
(
1
+
x
1
)
(
1
+
x
2
)
⋯
(
1
+
x
k
)
=
n
+
1
, prove that one can uniquely choose the corresponding masses
d
1
,
d
2
,
…
,
d
k
d_1,d_2,\ldots,d_k
d
1
,
d
2
,
…
,
d
k
with
1
≤
d
1
<
…
<
d
k
1\le d_1<\ldots<d_k
1
≤
d
1
<
…
<
d
k
in order for the obtained set of weights is perfect. (d) Determine all perfect sets of weights of total mass
1993
1993
1993
.