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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
2000 French Mathematical Olympiad
2000 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(3)
Exercise 2
1
Hide problems
existence of sphere, parameter
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be three distinct points in space,
(
A
)
(A)
(
A
)
the sphere with center
A
A
A
and radius
r
r
r
. Let
E
E
E
be the set of numbers
R
>
0
R>0
R
>
0
for which there is a sphere
(
H
)
(H)
(
H
)
with center
H
H
H
and radius
R
R
R
such that
B
B
B
and
C
C
C
are outside the sphere, and the points of the sphere
(
A
)
(A)
(
A
)
are strictly inside it.(a) Suppose that
B
B
B
and
C
C
C
are on a line with
A
A
A
and strictly outside
(
A
)
(A)
(
A
)
. Show that
E
E
E
is nonempty and bounded, and determine its supremum in terms of the given data. (b) Find a necessary and sufficient condition for
E
E
E
to be nonempty and bounded (c) Given
r
r
r
, compute the smallest possible supremum of
E
E
E
, if it exists.
Problem
1
Hide problems
cartesian triangles
In this problem we consider so-called cartesian triangles, that is, triangles
A
B
C
ABC
A
BC
with integer sides
B
C
=
a
,
C
A
=
b
,
A
B
=
c
BC=a,CA=b,AB=c
BC
=
a
,
C
A
=
b
,
A
B
=
c
and
∠
A
=
2
π
3
\angle A=\frac{2\pi}3
∠
A
=
3
2
π
. Unless noted otherwise,
△
A
B
C
\triangle ABC
△
A
BC
is assumed to be cartesian.(a) If
U
,
V
,
W
U,V,W
U
,
V
,
W
are the projections of the orthocenter
H
H
H
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively, specify which of the segments
A
U
AU
A
U
,
B
V
BV
B
V
,
C
W
CW
C
W
,
H
A
HA
H
A
,
H
B
HB
H
B
,
H
C
HC
H
C
,
H
U
HU
H
U
,
H
V
HV
H
V
,
H
W
HW
H
W
,
A
W
AW
A
W
,
A
V
AV
A
V
,
B
U
BU
B
U
,
B
W
BW
B
W
,
C
V
CV
C
V
,
C
U
CU
C
U
have rational length. (b) If
I
I
I
is the incenter,
J
J
J
the excenter across
A
A
A
, and
P
,
Q
P,Q
P
,
Q
the intersection points of the two bisectors at
A
A
A
with the line
B
C
BC
BC
, specify those of the segments
P
B
PB
PB
,
P
C
PC
PC
,
Q
B
QB
QB
,
Q
C
QC
QC
,
A
I
AI
A
I
,
A
J
AJ
A
J
,
A
P
AP
A
P
,
A
Q
AQ
A
Q
having rational length. (c) Assume that
b
b
b
and
c
c
c
are prime. Prove that exactly one of the numbers
a
+
b
−
c
a+b-c
a
+
b
−
c
and
a
−
b
+
c
a-b+c
a
−
b
+
c
is a multiple of
3
3
3
. (d) Assume that
a
+
b
−
c
3
c
=
p
q
\frac{a+b-c}{3c}=\frac pq
3
c
a
+
b
−
c
=
q
p
, where
p
p
p
and
q
q
q
are coprime, and denote by
d
d
d
the
gcd
\gcd
g
cd
of
p
(
3
p
+
2
q
)
p(3p+2q)
p
(
3
p
+
2
q
)
and
q
(
2
p
+
q
)
q(2p+q)
q
(
2
p
+
q
)
. Compute
a
,
b
,
c
a,b,c
a
,
b
,
c
in terms of
p
,
q
,
d
p,q,d
p
,
q
,
d
. (e) Prove that if
q
q
q
is not a multiple of
3
3
3
, then
d
=
1
d=1
d
=
1
. (f) Deduce a necessary and sufficient condition for a triangle to be cartesian with coprime integer sides, and by geometrical observations derive an analogous characterization of triangles
A
B
C
ABC
A
BC
with coprime sides
B
C
=
a
BC=a
BC
=
a
,
C
A
=
b
CA=b
C
A
=
b
,
A
B
=
c
AB=c
A
B
=
c
and
∠
A
=
π
3
\angle A=\frac\pi3
∠
A
=
3
π
.
Exercise 1
1
Hide problems
maximizing probability in balls and urns
We are given
b
b
b
white balls and
n
n
n
black balls (
b
,
n
>
0
b,n>0
b
,
n
>
0
) which are to be distributed among two urns, at least one in each. Let
s
s
s
be the number of balls in the first urn, and
r
r
r
the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it.(a) Compute the probability
p
p
p
that the drawn ball is white. (b) If
s
s
s
is fixed, for which
r
r
r
is
p
p
p
maximal? (c) Find the distribution of balls among the urns which maximizes
p
p
p
. (d) Give a generalization for larger numbers of colors and urns.