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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1985 Greece National Olympiad
1985 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
2
Hide problems
classic telescopic, f(x)=4^x/(4^x+2) - 1985 Greece MO X p4
Consider function
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
with
f
(
x
)
=
4
x
4
x
+
2
,
f(x)=\frac{4^x}{4^x+2},
f
(
x
)
=
4
x
+
2
4
x
,
for any
x
∈
R
x\in \mathbb{R}
x
∈
R
a) Prove that
f
(
x
)
+
f
(
1
−
x
)
=
1
,
f(x)+f(1-x)=1,
f
(
x
)
+
f
(
1
−
x
)
=
1
,
b) Claculate the sum
f
(
1
1986
)
+
f
(
2
1986
)
+
⋯
f
(
1986
1986
)
.
f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).
f
(
1986
1
)
+
f
(
1986
2
)
+
⋯
f
(
1986
1986
)
.
vector spaces problem from 1st Greek MO - 1985
Given the vector spaces
V
,
W
V,W
V
,
W
with coefficients over a field
K
K
K
and function
ϕ
:
V
→
W
\phi :V\to W
ϕ
:
V
→
W
satisfying the relation :
φ
(
λ
x
+
y
)
=
λ
φ
(
x
)
+
ϕ
(
y
)
\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)
φ
(
λ
x
+
y
)
=
λ
φ
(
x
)
+
ϕ
(
y
)
for all
x
,
y
∈
V
,
λ
∈
K
x,y \in V, \lambda \in K
x
,
y
∈
V
,
λ
∈
K
. Such a function is called linear.Let
L
φ
=
{
x
∈
V
/
φ
(
x
)
=
0
}
L\varphi=\{x\in V/\varphi(x)=0\}
L
φ
=
{
x
∈
V
/
φ
(
x
)
=
0
}
, and
M
=
φ
(
V
)
M=\varphi(V)
M
=
φ
(
V
)
, prove that :(i)
L
φ
L\varphi
L
φ
is subspace of
V
V
V
and
M
M
M
is subspace of
W
W
W
(ii)
L
φ
=
O
L\varphi={O}
L
φ
=
O
iff
φ
\varphi
φ
is
1
−
1
1-1
1
−
1
(iii) Dimension of
V
V
V
equals to dimension of
L
φ
L\varphi
L
φ
plus dimension of
M
M
M
(iv) If
θ
:
R
3
→
R
3
\theta : \mathbb{R}^3\to\mathbb{R}^3
θ
:
R
3
→
R
3
with
θ
(
x
,
y
,
z
)
=
(
2
x
−
z
,
x
−
y
,
x
−
3
y
+
z
)
\theta(x,y,z)=(2x-z,x-y,x-3y+z)
θ
(
x
,
y
,
z
)
=
(
2
x
−
z
,
x
−
y
,
x
−
3
y
+
z
)
, prove that
θ
\theta
θ
is linear function . Find
L
θ
=
{
x
∈
R
3
/
θ
(
x
)
=
0
}
L\theta=\{x\in {R}^3/\theta(x)=0\}
L
θ
=
{
x
∈
R
3
/
θ
(
x
)
=
0
}
and dimension of
M
=
θ
(
R
3
)
M=\theta({R}^3)
M
=
θ
(
R
3
)
.
1
2
Hide problems
1/ sin^2 b, 1/cos^2 b, integer roots of x^2-ax+a=0 - 1985 Greece MO X p1
Find all arcs
θ
\theta
θ
such that
1
sin
2
θ
,
1
cos
2
θ
\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta}
s
i
n
2
θ
1
,
c
o
s
2
θ
1
are integer numbers and roots of equation
x
2
−
a
x
+
a
=
0.
x^2-ax+a=0.
x
2
−
a
x
+
a
=
0.
E_A x{OA}+E_B x {OB}+E_C x{OC}=O - Carathéodory
Inside triangle
A
B
C
ABC
A
BC
consider random point
O
O
O
. Prove that:
E
A
O
A
→
+
E
B
O
B
→
+
E
C
O
C
→
=
O
→
E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}
E
A
O
A
+
E
B
OB
+
E
C
OC
=
O
where
E
A
,
E
B
,
E
C
E_A,E_B,E_C
E
A
,
E
B
,
E
C
the areas of triangle
B
O
C
,
C
O
B
,
A
O
B
BOC, COB, AOB
BOC
,
COB
,
A
OB
respectively
3
2
Hide problems
lake visibilty problem - 1985 Greece MO X p3
Interior in alake there are two points
A
,
B
A,B
A
,
B
from which we can see every other point of the lake. Prove that also from any other point of the segment
A
B
AB
A
B
, we can see all points of the lake.
min distance of lattice point from 5x-10y+3=0 is \sqrt3/ 20
Consider the line (E):
5
x
−
10
y
+
3
=
0
5x-10y+3=0
5
x
−
10
y
+
3
=
0
. Prove that: a) Line
(
E
)
(E)
(
E
)
doesn't pass through points with integer coordinates. b) There is no point
A
(
a
1
,
a
2
)
A(a_1,a_2)
A
(
a
1
,
a
2
)
with
a
1
,
a
2
∈
Z
a_1,a_2 \in \mathbb{Z}
a
1
,
a
2
∈
Z
with distance from
(
E
)
(E)
(
E
)
less then
3
20
\frac{\sqrt3}{20}
20
3
.
2
2
Hide problems
3 acute angles is max for convex n-gon, equilateral 1985 Greece MO X p2
a) Prove that a convex
n
n
n
-gon cannot have more than
3
3
3
interior angles acute. b) Prove that a convex
n
n
n
-gon that has
3
3
3
interior angles equal to
6
0
0
,
60^0,
6
0
0
,
is equilateral.
f(x)=x has real solution(s) if f(f(x))=x has real solution(s), f continuous
Conside the continuous
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
. It is also know that equation
f
(
f
(
f
(
x
)
)
)
=
x
f(f(f(x)))=x
f
(
f
(
f
(
x
)))
=
x
has solution in
R
\mathbb{R}
R
. Prove that equation
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
has solution in
R
\mathbb{R}
R
.