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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
2003 French Mathematical Olympiad
2003 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(1)
1
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5-part geometry problem, orthocentric sets
Preliminary Part (a) Let
M
M
M
be a point in the plane of a triangle
A
B
C
ABC
A
BC
. Prove that
M
A
→
⋅
B
C
→
+
M
B
→
⋅
C
A
→
+
M
C
→
⋅
A
B
→
=
0
\overrightarrow{MA}\cdot\overrightarrow{BC}+\overrightarrow{MB}\cdot\overrightarrow{CA}+\overrightarrow{MC}\cdot\overrightarrow{AB}=0
M
A
⋅
BC
+
MB
⋅
C
A
+
MC
⋅
A
B
=
0
Deduce that the altitudes of triangle
A
B
C
ABC
A
BC
are concurrent at a point
H
H
H
, called the orthocenter.(b) Let
Ω
\Omega
Ω
be the circumcenter of a triangle
A
B
C
ABC
A
BC
and
H
H
H
be the point given by
Ω
H
→
=
Ω
A
→
+
Ω
B
→
+
Ω
C
→
\overrightarrow{\Omega H}=\overrightarrow{\Omega A}+\overrightarrow{\Omega B}+\overrightarrow{\Omega C}
Ω
H
=
Ω
A
+
Ω
B
+
Ω
C
. Show that
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
. For every set
X
X
X
of points in the plane, we denote by
H
(
X
)
\mathcal H(X)
H
(
X
)
the set of orthocenters of all triangles with vertices in
X
X
X
. We call a planar set
X
X
X
orthocentric if it does not contain any line and contains the entire
H
(
X
)
\mathcal H(X)
H
(
X
)
. [hide=Part 1] (a) Find all orthocentric sets of three points. (b) Find all orthocentric sets of four points. (c) Let
X
X
X
be a set of four points on a circle and let
Y
=
H
(
x
)
Y=\mathcal H(x)
Y
=
H
(
x
)
. i. Prove that
Y
Y
Y
is the image of
X
X
X
under an isometry. ii. Determine
H
(
Y
)
\mathcal H(Y)
H
(
Y
)
. (d) i. If
Γ
\Gamma
Γ
is a nondegenerate circle, determine
H
(
Γ
)
\mathcal H(\Gamma)
H
(
Γ
)
. ii. If
D
D
D
is a nondegenerate disk, determine
H
(
D
)
\mathcal H(D)
H
(
D
)
.[hide=Part 2] In this part,
R
R
R
is a positive number,
n
n
n
an integer not smaller than
2
2
2
, and
X
X
X
the set of vertices of a regular
2
n
2n
2
n
-gon inscribed in the circle with center
O
O
O
and radius
R
R
R
. Consider the set
T
\mathfrak T
T
of triangles with the vertices in
X
X
X
. An element of
T
\mathfrak T
T
is chosen at random.(a) What is the probability of choosing a right-angled triangle? (b) What is the probability of choosing an acute triangle? (c) Let
L
L
L
be the squared distance from
O
O
O
to the orthocenter of the chosen triangle. Find the expected value of
L
L
L
.[hide=Part 3] (a) Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers with
a
(
b
−
c
)
≠
0
a(b-c)\ne0
a
(
b
−
c
)
=
0
and
A
,
B
,
C
A,B,C
A
,
B
,
C
be the points
(
0
,
a
)
(0,a)
(
0
,
a
)
,
(
b
,
0
)
(b,0)
(
b
,
0
)
,
(
c
,
0
)
(c,0)
(
c
,
0
)
, respectively. Compute the coordinates of the orthocenter
D
D
D
of
△
A
B
C
\triangle ABC
△
A
BC
. (b) Let
X
X
X
be the union of a line
δ
\delta
δ
and a point
M
M
M
outside
δ
\delta
δ
. Determine
H
(
X
)
\mathcal H(X)
H
(
X
)
. Prove that
H
(
x
)
∪
X
\mathcal H(x)\cup X
H
(
x
)
∪
X
is an orthocentric set. (c) Let
X
X
X
be an orthocentric set contained in the union of the coordinate axes
x
x
x
and
y
y
y
and containing at least three points distinct from
O
O
O
. i. Show that
X
X
X
contains at least three points on axis
y
y
y
with nonzero
x
x
x
-coordinates of the same sign. ii. Show that
X
X
X
contains at least three points on axis
y
y
y
with positive
x
x
x
-coordinates. (d) i. Find all finite orthocentric sets of at most five points contained in the union of the axes
x
x
x
and
y
y
y
. ii. Let
X
X
X
be an orthocentric set contained in the union of the axes
x
x
x
and
y
y
y
and consisting of at least six points. Prove that there exist two sequences
(
x
n
)
(x_n)
(
x
n
)
,
(
x
’
n
)
(x’_n)
(
x
’
n
)
of nonzero real numbers such that, for each
n
n
n
, the points
(
x
n
,
0
)
(x_n,0)
(
x
n
,
0
)
and
(
x
’
n
,
0
)
(x’_n,0)
(
x
’
n
,
0
)
are in
X
X
X
, and
lim
n
→
∞
=
∞
,
lim
n
→
∞
x
’
n
=
0.
\lim_{n\to\infty}=\infty,\qquad\lim_{n\to\infty}x’^n=0.
n
→
∞
lim
=
∞
,
n
→
∞
lim
x
’
n
=
0.
Can the set
X
X
X
be finite?[hide=Part 4] In this part we are concerned with constructing some remarkable orthocentric sets. (a) Let
k
k
k
be a nonzero real number and
Y
Y
Y
be the hyperbola
x
y
=
k
xy=k
x
y
=
k
. i. Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be distinct points of
Y
Y
Y
, with the respective abscissas
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
. Prove that
A
B
AB
A
B
and
C
D
CD
C
D
are orthogonal if and only if
a
b
c
d
=
−
k
2
abcd=-k^2
ab
c
d
=
−
k
2
. ii. With the above notation, find the orthocenter of triangle
A
B
C
ABC
A
BC
. iii. Prove that
Y
Y
Y
is orthocentric.Throughout this part,
q
q
q
denotes a nonzero integer and
X
X
X
the set of points
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfying
x
2
+
q
x
y
−
y
2
=
1
x^2+qxy-y^2=1
x
2
+
q
x
y
−
y
2
=
1
.(a) i. Prove that the equation
t
2
−
q
t
−
1
t^2-qt-1
t
2
−
qt
−
1
has two distinct real roots and show that these roots are irrational. Throughout this part,
r
r
r
and
r
’
r’
r
’
denote these roots and
s
s
s
the similitude defined by
z
↦
(
1
−
r
i
)
z
z\mapsto(1-ri)z
z
↦
(
1
−
r
i
)
z
in terms of complex numbers. i. Prove that
s
(
X
)
s(X)
s
(
X
)
is the hyperbola given by
x
y
=
k
xy = k
x
y
=
k
for some real
k
k
k
, and find
k
k
k
. Deduce that
X
X
X
is an orthocentric set.(b) Let
G
G
G
be the set of integer points of
X
X
X
and
Γ
\Gamma
Γ
be the set of
x
x
x
-coordinates of the elements of
s
(
G
)
s(G)
s
(
G
)
. i. Show that
Γ
\Gamma
Γ
is the set of numbers of the form
x
+
r
y
x+ry
x
+
ry
, where
x
,
y
x,y
x
,
y
are integers and
(
x
+
r
y
)
(
x
+
r
’
y
)
=
1
(x+ry)(x+r’y)=1
(
x
+
ry
)
(
x
+
r
’
y
)
=
1
. ii. Show that
−
1
∈
Γ
-1\in\Gamma
−
1
∈
Γ
and
r
2
∈
Γ
r^2\in\Gamma
r
2
∈
Γ
. iii. Prove that the product of two elements in
Γ
\Gamma
Γ
and the inverse of an element of
Γ
\Gamma
Γ
are in
Γ
\Gamma
Γ
. Show that
Γ
\Gamma
Γ
is infinite.(c) Conclude that the set
G
G
G
of integer points of
X
X
X
is an infinite orthocentric set.