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Problems
Contests
National and Regional Contests
France Contests
French Mathematical Olympiad
1986 French Mathematical Olympiad
1986 French Mathematical Olympiad
Part of
French Mathematical Olympiad
Subcontests
(5)
Problem 5
1
Hide problems
iterating function f(x)=sqrt4(1-x)
The functions
f
,
g
:
[
0
,
1
]
→
R
f,g:[0,1]\to\mathbb R
f
,
g
:
[
0
,
1
]
→
R
are given with the formulas
f
(
x
)
=
1
−
x
4
,
g
(
x
)
=
f
(
f
(
x
)
)
,
f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),
f
(
x
)
=
4
1
−
x
,
g
(
x
)
=
f
(
f
(
x
))
,
and
c
c
c
denotes any solution of
x
=
f
(
x
)
x=f(x)
x
=
f
(
x
)
.(a) i. Analyze the function
f
(
x
)
f(x)
f
(
x
)
and draw its graph. Prove that the equation
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
has the unique root
c
c
c
satisfying
c
∈
[
0.72
,
0.73
]
c\in[0.72,0.73]
c
∈
[
0.72
,
0.73
]
. ii. Analyze the function
f
′
(
x
)
f'(x)
f
′
(
x
)
. Let
M
1
M_1
M
1
and
M
2
M_2
M
2
be the points of the graph of
f
(
x
)
f(x)
f
(
x
)
with different
x
x
x
coordinates. What is the position of the arc
M
1
M
2
M_1M_2
M
1
M
2
of the graph with respect to the segment
M
1
M
2
M_1M_2
M
1
M
2
? iii. Analyze the function
g
(
x
)
g(x)
g
(
x
)
and draw its graph. What is the position of that graph with respect to the line
y
=
x
y=x
y
=
x
? Find the tangents to the graph at points with
x
x
x
coordinates
0
0
0
and
1
1
1
. iv. Prove that every sequence
{
a
n
}
\{a_n\}
{
a
n
}
with the conditions
a
1
∈
(
0
,
1
)
a_1\in(0,1)
a
1
∈
(
0
,
1
)
and
a
n
+
1
=
f
(
a
n
)
a_{n+1}=f(a_n)
a
n
+
1
=
f
(
a
n
)
for
n
∈
N
n\in\mathbb N
n
∈
N
converges. [hide=Official Hint]Consider the sequences
{
a
2
n
−
1
}
,
{
a
2
n
}
(
n
∈
N
)
\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)
{
a
2
n
−
1
}
,
{
a
2
n
}
(
n
∈
N
)
and the function
g
(
x
)
g(x)
g
(
x
)
associated with the graph. (b) On the graph of the function
f
(
x
)
f(x)
f
(
x
)
consider the points
M
M
M
and
M
′
M'
M
′
with
x
x
x
coordinates
x
x
x
and
f
(
x
)
f(x)
f
(
x
)
, where
x
≠
c
x\ne c
x
=
c
. i. Prove that the line
M
M
′
MM'
M
M
′
intersects with the line
y
=
x
y=x
y
=
x
at the point with
x
x
x
coordinate
h
(
x
)
=
x
−
(
f
(
x
)
−
x
)
2
g
(
x
)
+
x
−
2
f
(
x
)
.
h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.
h
(
x
)
=
x
−
g
(
x
)
+
x
−
2
f
(
x
)
(
f
(
x
)
−
x
)
2
.
ii. Prove that if
x
∈
(
0
,
c
)
x\in(0,c)
x
∈
(
0
,
c
)
then
h
(
x
)
∈
(
x
,
c
)
h(x)\in(x,c)
h
(
x
)
∈
(
x
,
c
)
. iii. Analyze whether the sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfying
a
1
∈
(
0
,
c
)
,
a
n
+
1
=
h
(
a
n
)
a_1\in(0,c),a_{n+1}=h(a_n)
a
1
∈
(
0
,
c
)
,
a
n
+
1
=
h
(
a
n
)
for
n
∈
N
n\in\mathbb N
n
∈
N
converges. Prove that the sequence
{
a
n
+
1
−
c
a
n
−
c
}
\{\tfrac{a_{n+1}-c}{a_n-c}\}
{
a
n
−
c
a
n
+
1
−
c
}
converges and find its limit. (c) Assume that the calculator approximates every number
b
∈
[
−
2
,
2
]
b\in[-2,2]
b
∈
[
−
2
,
2
]
by number
b
‾
\overline b
b
having
p
p
p
decimal digits after the decimal point. We are performing the following sequence of operations on that calculator:1) Set
a
=
0.72
a=0.72
a
=
0.72
; 2) Calculate
δ
(
a
)
=
f
(
a
)
‾
−
a
\delta(a)=\overline{f(a)}-a
δ
(
a
)
=
f
(
a
)
−
a
; 3) If
∣
δ
(
a
)
∣
>
0.5
⋅
1
0
−
p
|\delta(a)|>0.5\cdot10^{-p}
∣
δ
(
a
)
∣
>
0.5
⋅
1
0
−
p
, then calculate
h
(
a
)
‾
\overline{h(a)}
h
(
a
)
and go to the operation
2
)
2)
2
)
using
h
(
a
)
‾
\overline{h(a)}
h
(
a
)
instead of
a
a
a
; 4) If
∣
δ
(
a
)
∣
≤
0.5
⋅
1
0
−
p
|\delta(a)|\le0.5\cdot10^{-p}
∣
δ
(
a
)
∣
≤
0.5
⋅
1
0
−
p
, finish the calculation.Let
c
↔
\overleftrightarrow c
c
be the last of calculated values for
h
(
a
)
‾
\overline{h(a)}
h
(
a
)
. Assuming that for each
x
∈
[
0.72
,
0.73
]
x\in[0.72,0.73]
x
∈
[
0.72
,
0.73
]
we have
∣
f
(
x
)
‾
−
f
(
x
)
∣
<
ϵ
\left|\overline{f(x)}-f(x)\right|<\epsilon
f
(
x
)
−
f
(
x
)
<
ϵ
, determine
δ
(
c
↔
)
\delta(\overleftrightarrow c)
δ
(
c
)
, the accuracy (depending on
ϵ
\epsilon
ϵ
) of the approximation of
c
c
c
with
c
↔
\overleftrightarrow c
c
. (d) Assume that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
=
0.72
a_1=0.72
a
1
=
0.72
and
a
n
+
1
=
f
(
a
n
)
a_{n+1}=f(a_n)
a
n
+
1
=
f
(
a
n
)
for
n
∈
N
n\in\mathbb N
n
∈
N
. Find the smallest
n
0
∈
N
n_0\in\mathbb N
n
0
∈
N
, such that for every
n
≥
n
0
n\ge n_0
n
≥
n
0
we have
∣
a
n
−
c
∣
<
1
0
−
6
|a_n-c|<10^{-6}
∣
a
n
−
c
∣
<
1
0
−
6
.
Problem 4
1
Hide problems
1&2 finite differences of sequence
For every sequence
{
a
n
}
(
n
∈
N
)
\{a_n\}~(n\in\mathbb N)
{
a
n
}
(
n
∈
N
)
we define the sequences
{
Δ
a
n
}
\{\Delta a_n\}
{
Δ
a
n
}
and
{
Δ
2
a
n
}
\{\Delta^2a_n\}
{
Δ
2
a
n
}
by the following formulas: \begin{align*}\Delta a_n&=a_{n+1}-a_n,\\\Delta^2a_n&=\Delta a_{n+1}-\Delta a_n.\end{align*}Further, for all
n
∈
N
n\in\mathbb N
n
∈
N
for which
Δ
a
n
2
≠
0
\Delta a_n^2\ne0
Δ
a
n
2
=
0
, define
a
n
′
=
a
n
−
(
Δ
a
n
)
2
Δ
2
a
n
.
a_n'=a_n-\frac{(\Delta a_n)^2}{\Delta^2a_n}.
a
n
′
=
a
n
−
Δ
2
a
n
(
Δ
a
n
)
2
.
(a) For which sequences
{
a
n
}
\{a_n\}
{
a
n
}
is the sequence
{
Δ
2
a
n
}
\{\Delta^2a_n\}
{
Δ
2
a
n
}
constant? (b) Find all sequences
{
a
n
}
\{a_n\}
{
a
n
}
, for which the numbers
a
n
′
a_n'
a
n
′
are defined for all
n
∈
N
n\in\mathbb N
n
∈
N
and for which the sequence
{
a
n
′
}
\{a_n'\}
{
a
n
′
}
is constant. (c) Assume that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
converges to
a
=
0
a=0
a
=
0
, and
a
n
≠
a
a_n\ne a
a
n
=
a
for all
n
∈
N
n\in\mathbb N
n
∈
N
and the sequence
{
a
n
+
1
−
a
a
n
−
a
}
\{\tfrac{a_{n+1}-a}{a_n-a}\}
{
a
n
−
a
a
n
+
1
−
a
}
converges to
λ
≠
1
\lambda\ne1
λ
=
1
. i. Prove that
λ
∈
[
−
1
,
1
)
\lambda\in[-1,1)
λ
∈
[
−
1
,
1
)
. ii. Prove that there exists
n
0
∈
N
n_0\in\mathbb N
n
0
∈
N
such that for all integers
n
≥
n
0
n\ge n_0
n
≥
n
0
we have
Δ
2
a
n
≠
0
\Delta^2a_n\ne0
Δ
2
a
n
=
0
. iii. Let
λ
≠
0
\lambda\ne0
λ
=
0
. For which
k
∈
Z
k\in\mathbb Z
k
∈
Z
is the sequence
{
a
n
′
a
n
+
k
}
\{\tfrac{a_n'}{a_{n+k}}\}
{
a
n
+
k
a
n
′
}
not convergent? iv. Let
λ
=
0
\lambda=0
λ
=
0
. Prove that the sequences
{
a
n
′
/
a
n
}
\{a_n'/a_n\}
{
a
n
′
/
a
n
}
and
{
a
n
′
/
a
n
+
1
}
\{a_n'/a_{n+1}\}
{
a
n
′
/
a
n
+
1
}
converge to
0
0
0
. Find an example of
{
a
n
}
\{a_n\}
{
a
n
}
for which the sequence
{
a
n
′
/
a
n
+
2
}
\{a_n'/a_{n+2}\}
{
a
n
′
/
a
n
+
2
}
has a non-zero limit. (d) What happens with part (c) if we remove the condition
a
=
0
a=0
a
=
0
?
Problem 3
1
Hide problems
complex inequalities, |z|+|w|≤|z+w|+|z-w|
(a) Prove or find a counter-example: For every two complex numbers
z
,
w
z,w
z
,
w
the following inequality holds:
∣
z
∣
+
∣
w
∣
≤
∣
z
+
w
∣
+
∣
z
−
w
∣
.
|z|+|w|\le|z+w|+|z-w|.
∣
z
∣
+
∣
w
∣
≤
∣
z
+
w
∣
+
∣
z
−
w
∣.
(b) Prove that for all
z
1
,
z
2
,
z
3
,
z
4
∈
C
z_1,z_2,z_3,z_4\in\mathbb C
z
1
,
z
2
,
z
3
,
z
4
∈
C
:
∑
k
=
1
4
∣
z
k
∣
≤
∑
1
≤
i
<
j
≤
4
∣
z
i
+
z
j
∣
.
\sum_{k=1}^4|z_k|\le\sum_{1\le i<j\le4}|z_i+z_j|.
k
=
1
∑
4
∣
z
k
∣
≤
1
≤
i
<
j
≤
4
∑
∣
z
i
+
z
j
∣.
Problem 2
1
Hide problems
inequalities on four points in a plane, existence
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
, and
M
M
M
are given in the plane. (a) Let
D
D
D
be the point in the plane such that
D
A
≤
C
A
DA\le CA
D
A
≤
C
A
and
D
B
≤
C
B
DB\le CB
D
B
≤
CB
. Prove that there exists point
N
N
N
satisfying
N
A
≤
M
A
,
N
B
≤
M
B
NA\le MA,NB\le MB
N
A
≤
M
A
,
NB
≤
MB
, and
N
D
≤
M
C
ND\le MC
N
D
≤
MC
. (b) Let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the points in the plane such that
A
′
B
′
≤
A
B
,
A
′
C
′
≤
A
C
,
B
′
C
′
≤
B
C
A'B'\le AB,A'C'\le AC,B'C'\le BC
A
′
B
′
≤
A
B
,
A
′
C
′
≤
A
C
,
B
′
C
′
≤
BC
. Does there exist a point
M
′
M'
M
′
which satisfies the inequalities
M
′
A
′
≤
M
A
,
M
′
B
′
≤
M
B
,
M
′
C
′
≤
M
C
M'A'\le MA,M'B'\le MB,M'C'\le MC
M
′
A
′
≤
M
A
,
M
′
B
′
≤
MB
,
M
′
C
′
≤
MC
?
Problem 1
1
Hide problems
midpoints of tetrahedron edges coplanar
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron. (a) Prove that the midpoints of the edges
A
B
,
A
C
,
B
D
AB,AC,BD
A
B
,
A
C
,
B
D
, and
C
D
CD
C
D
lie in a plane. (b) Find the point in that plane, whose sum of distances from the lines
A
D
AD
A
D
and
B
C
BC
BC
is minimal.