MathDB
Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
2002 French Mathematical Olympiad
2002 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(1)
1
Hide problems
pseudo-right triangles, locus x^2+y^2=z^2+1
[hide=Part 1] For a triangle
A
B
C
ABC
A
BC
, we denote by
P
P
P
the orthogonal projection of
A
A
A
on
B
C
BC
BC
and by
D
D
D
the reflection of
C
C
C
in
A
P
AP
A
P
.Triangle
A
B
C
ABC
A
BC
is said to be pseudo-right at
A
A
A
if
∣
∠
B
−
∠
C
∣
=
π
2
\left|\angle B-\angle C\right|=\frac\pi2
∣
∠
B
−
∠
C
∣
=
2
π
. Specifically, it is pseudo-right at
A
A
A
and obtuse at
B
B
B
if
∠
B
−
∠
C
=
π
2
\angle B-\angle C=\frac\pi2
∠
B
−
∠
C
=
2
π
.(a) Prove that triangle
A
B
C
ABC
A
BC
is pseudo-right at
A
A
A
if and only if triangle
A
B
D
ABD
A
B
D
is right at
A
A
A
. (b) Prove that
P
A
2
=
P
B
⋅
P
C
PA^2=PB\cdot PC
P
A
2
=
PB
⋅
PC
if and only if
△
A
B
C
\triangle ABC
△
A
BC
is right at
A
A
A
or pseudo-right at
A
A
A
. (c) Prove that triangle
A
B
C
ABC
A
BC
is pseudo-right at
A
A
A
if and only if its orthocenter is symmetric to
A
A
A
with respect to
B
C
BC
BC
. (d) Let
R
R
R
be the circumradius of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that
P
B
+
P
C
=
2
R
PB+PC=2R
PB
+
PC
=
2
R
if and only if
△
A
B
C
\triangle ABC
△
A
BC
is right at
A
A
A
or pseudo-right at
A
A
A
. (e) Prove that
△
A
B
C
\triangle ABC
△
A
BC
is pseudo-right at
A
A
A
if and only if the line
A
P
AP
A
P
is tangent to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
. (f) Let
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
be the points in the complex plane corresponding to
A
,
B
,
C
,
A,B,C,
A
,
B
,
C
,
respectively. i. Give a necessary and sufficient condition on
α
−
β
α
−
γ
(
β
−
γ
)
2
\frac{\alpha-\beta}{\alpha-\gamma}(\beta-\gamma)^2
α
−
γ
α
−
β
(
β
−
γ
)
2
that
△
A
B
C
\triangle ABC
△
A
BC
is pseudo-right at
A
A
A
. ii. Set
β
=
−
γ
=
e
i
π
4
\beta=-\gamma=e^{\frac{i\pi}4}
β
=
−
γ
=
e
4
iπ
. Find the set
E
1
E_1
E
1
of points
A
A
A
in the plane for which
△
A
B
C
\triangle ABC
△
A
BC
is pseudo-right at
A
A
A
. iii. Set
β
=
−
γ
=
1
\beta=-\gamma=1
β
=
−
γ
=
1
. Find the set
E
2
E_2
E
2
of points
A
A
A
in the plane for which
△
A
B
C
\triangle ABC
△
A
BC
is pseudo-right at
A
A
A
. iv. Which geometric transformation takes
E
2
E_2
E
2
to
E
1
E_1
E
1
?[hide=Part 2] (a) Let
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
be a triple of positive numbers. Prove that the following conditions are equivalent: (i) There is a pseudo-right at
A
A
A
and obtuse at
B
B
B
triangle
A
B
C
ABC
A
BC
with
A
B
=
c
AB=c
A
B
=
c
,
B
C
=
a
BC=a
BC
=
a
,
C
A
=
b
CA=b
C
A
=
b
. (ii)
b
2
−
c
2
=
a
b
2
+
c
2
b^2-c^2=a\sqrt{b^2+c^2}
b
2
−
c
2
=
a
b
2
+
c
2
(iii) There exist real numbers
ρ
>
0
\rho>0
ρ
>
0
and
0
<
θ
<
π
4
0<\theta<\frac\pi4
0
<
θ
<
4
π
such that
a
=
ρ
cos
2
θ
a=\rho\cos2\theta
a
=
ρ
cos
2
θ
,
b
=
ρ
cos
θ
b=\rho\cos\theta
b
=
ρ
cos
θ
, and
c
=
ρ
sin
θ
c=\rho\sin\theta
c
=
ρ
sin
θ
.If these conditions are satisfied, prove that
θ
\theta
θ
is the measure of one of the angles of
∠
A
B
C
\angle ABC
∠
A
BC
. Can you give a geometric interpretation for
ρ
\rho
ρ
? (b) Let
△
A
B
C
\triangle ABC
△
A
BC
be pseudo-right at
A
A
A
and obtuse at
B
B
B
and let its side lengths be rational. Define
ρ
\rho
ρ
and
θ
\theta
θ
as above. In this question you can use without proof that
cos
2
ϕ
=
1
−
tan
2
ϕ
1
+
tan
2
ϕ
\cos2\phi=\frac{1-\tan^2\phi}{1+\tan^2\phi}
cos
2
ϕ
=
1
+
t
a
n
2
ϕ
1
−
t
a
n
2
ϕ
and
sin
2
ϕ
=
2
tan
ϕ
1
+
tan
2
ϕ
\sin2\phi=\frac{2\tan\phi}{1+\tan^2\phi}
sin
2
ϕ
=
1
+
t
a
n
2
ϕ
2
t
a
n
ϕ
. i. Prove that
ρ
\rho
ρ
is rational and deduce that so in
tan
θ
2
\tan\frac\theta2
tan
2
θ
. Let
p
,
q
p,q
p
,
q
be the coprime positive integers with
tan
θ
2
=
p
q
\tan\frac\theta2=\frac pq
tan
2
θ
=
q
p
. ii. Prove that
0
<
p
<
q
(
2
−
1
)
0<p<q\left(\sqrt2-1\right)
0
<
p
<
q
(
2
−
1
)
and show the existence of a positive rational number
r
r
r
such that
a
=
r
(
p
4
−
6
p
2
q
2
+
q
4
)
,
b
=
r
(
q
4
−
r
4
)
,
c
=
2
p
q
r
(
p
2
+
q
2
)
.
a=r\left(p^4-6p^2q^2+q^4\right),\qquad b=r\left(q^4-r^4\right),\qquad c=2pqr\left(p^2+q^2\right).
a
=
r
(
p
4
−
6
p
2
q
2
+
q
4
)
,
b
=
r
(
q
4
−
r
4
)
,
c
=
2
pq
r
(
p
2
+
q
2
)
.
(c) Conversely, show that the formulas in 2b give side lengths of a triangle that is pseudo-right at
A
A
A
and obtuse at
B
B
B
. (d) i. Let
p
p
p
and
q
q
q
be coprime positive integers. Find the greatest positive divisor of
p
4
−
6
p
2
q
2
+
q
4
,
q
4
−
p
4
,
2
p
q
(
p
2
+
q
2
)
p^4-6p^2q^2+q^4,q^4-p^4,2pq\left(p^2+q^2\right)
p
4
−
6
p
2
q
2
+
q
4
,
q
4
−
p
4
,
2
pq
(
p
2
+
q
2
)
in terms of parity of
p
p
p
and
q
q
q
. ii. Describe all triples of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
for which there is a triangle
A
B
C
ABC
A
BC
, pseudo-right at
A
A
A
and obtuse at
B
B
B
, with
A
B
=
c
AB=c
A
B
=
c
,
B
C
=
a
BC=a
BC
=
a
,
C
A
=
b
CA=b
C
A
=
b
. (e) Solve in
N
\mathbb N
N
the equation
x
2
(
y
2
+
z
2
)
=
(
y
2
−
z
2
)
2
x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2
x
2
(
y
2
+
z
2
)
=
(
y
2
−
z
2
)
2
. (f) Solve in
Q
+
\mathbb Q^+
Q
+
the equation
x
2
(
y
2
+
z
2
)
=
(
y
2
−
z
2
)
2
x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2
x
2
(
y
2
+
z
2
)
=
(
y
2
−
z
2
)
2
. (g) Solve in
N
\mathbb N
N
the equation
x
2
(
y
2
−
z
2
)
2
=
(
y
2
+
z
2
)
3
x^2\left(y^2-z^2\right)^2=\left(y^2+z^2\right)^3
x
2
(
y
2
−
z
2
)
2
=
(
y
2
+
z
2
)
3
.[hide=Part 3] Let
H
\mathcal H
H
be the curve defined by
x
≥
1
x\ge1
x
≥
1
and
y
=
x
2
−
1
y=\sqrt{x^2-1}
y
=
x
2
−
1
and let
A
=
(
r
,
s
)
A=(r,s)
A
=
(
r
,
s
)
be a point on
H
\mathcal H
H
. Denote by
A
\mathcal A
A
the area of the set of points satisfying
1
≤
x
≤
r
1\le x\le r
1
≤
x
≤
r
and
y
2
≤
x
2
−
1
y^2\le x^2-1
y
2
≤
x
2
−
1
.(a) Calculate
A
\mathcal A
A
in terms of
r
r
r
and
s
s
s
. (For example, you can rotate the image by
π
4
\frac\pi4
4
π
.) (b) (Based on a result by Pierre Fermat in 1658.) Let
u
u
u
be a positive and
n
n
n
be a natural number such that
u
n
=
r
+
s
u^n=r+s
u
n
=
r
+
s
. For each integer
k
k
k
,
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
, consider the right-angled trapezoid (possibly degenerated into a triangle) having a lateral side with endpoints at
(
u
k
−
1
,
0
)
\left(u^{k-1},0\right)
(
u
k
−
1
,
0
)
and
(
u
k
,
0
)
\left(u^k,0\right)
(
u
k
,
0
)
, the bases with slope
−
1
-1
−
1
, and the top right angle at the point on
H
\mathcal H
H
with abscissa
u
k
−
1
+
u
1
−
k
2
\frac{u^{k-1}+u^{1-k}}2
2
u
k
−
1
+
u
1
−
k
. i. Prove that the trapezoid
T
k
T_k
T
k
is well-defined for each
k
k
k
and draw a sketch. ii. Why can we conjecture that the sum of these areas of these trapezoids has the limit
A
+
s
2
2
\frac{\mathcal A+s^2}2
2
A
+
s
2
when
u
u
u
approaches
+
∞
+\infty
+
∞
? iii. Prove the conjecture using another sequence of trapezoids combined with the first. iv. Find the value of
A
\mathcal A
A
. (c) Let
B
=
(
1
,
0
)
B=(1,0)
B
=
(
1
,
0
)
and
C
=
(
−
1
,
0
)
C=(-1,0)
C
=
(
−
1
,
0
)
and let
A
=
(
x
,
y
)
A=(x,y)
A
=
(
x
,
y
)
be a point with
x
,
y
≥
0
x,y\ge0
x
,
y
≥
0
for which
△
A
B
C
\triangle ABC
△
A
BC
is a pseudo-rectangle at
A
A
A
. Denote by
S
S
S
the area of
△
A
B
C
\triangle ABC
△
A
BC
and by
S
’
S’
S
’
the area of the part of the triangle consisting of points
(
X
,
Y
)
(X,Y)
(
X
,
Y
)
with
Y
2
≤
X
2
−
1
Y^2\le X^2-1
Y
2
≤
X
2
−
1
. Determine, if it exists, the limit of
S
’
S
\frac{S’}S
S
S
’
when
x
→
∞
x\to\infty
x
→
∞
.[hide=Part 4] In the plane
z
=
0
z=0
z
=
0
in coordinate space, let
L
\mathfrak L
L
be the circle with center
O
O
O
and radius
1
1
1
and let
T
T
T
and
P
P
P
be distinct points such that
T
P
TP
TP
is the tangent to
L
\mathfrak L
L
at
T
T
T
. The line
O
P
OP
OP
meets
L
\mathfrak L
L
at
B
B
B
and
C
C
C
, and
D
\mathfrak D
D
is the line through
P
P
P
perpendicular to the plane
z
=
0
z=0
z
=
0
.(a) i. Show that there exist two points
A
,
A
’
A,A’
A
,
A
’
on
D
\mathfrak D
D
such that triangles
A
B
C
ABC
A
BC
and
A
’
B
C
A’BC
A
’
BC
are pseudo-right at
A
A
A
and
A
’
A’
A
’
. Show how to construct these points. ii. Prove that the coordinates of these two points satisfy
x
2
+
y
2
=
z
2
+
1
x^2+y^2=z^2+1
x
2
+
y
2
=
z
2
+
1
. (b) Let
H
\mathcal H
H
be the set of points
A
A
A
and
A
’
A’
A
’
when
T
T
T
and
P
P
P
vary. i. What is the intersection of
H
\mathcal H
H
with a plane orthogonal to the
x
x
x
-axis? ii. What is the intersection of
H
\mathcal H
H
with a plane containing the
x
x
x
-axis? iii. Prove that
H
\mathcal H
H
is a union of lines and describe these lines. (c) We are now interested in points of set
H
\mathcal H
H
with integer coordinates. i. Let
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
be one such point. Prove that
x
x
x
or
y
y
y
is odd. Denote by
I
\mathfrak I
I
the set of points
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
with positive integer coordinates and with
x
x
x
odd such that
x
2
+
y
2
=
z
2
+
1
x^2+y^2=z^2+1
x
2
+
y
2
=
z
2
+
1
. ii. Let
d
d
d
be a fixed positive integer. Prove that the set of points
(
x
,
y
,
z
)
∈
I
(x,y,z)\in\mathfrak I
(
x
,
y
,
z
)
∈
I
with
gcd
(
x
+
1
,
y
+
z
)
=
d
\gcd(x+1,y+z)=d
g
cd
(
x
+
1
,
y
+
z
)
=
d
is empty if
d
d
d
is odd and infinite if
d
d
d
is even. iii. Let
m
≥
3
m\ge3
m
≥
3
be an integer. How many elements
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of
I
\mathfrak I
I
with
x
=
m
x=m
x
=
m
are there? Write down these elements for
m
=
3
,
5
,
7
,
9
m=3,5,7,9
m
=
3
,
5
,
7
,
9
.