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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1990 Greece National Olympiad
1990 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
3
Hide problems
equailteral of <CAD=<CBE=30^o - 1990 Greece MO Grade X p4
In a triangle
A
B
C
ABC
A
BC
with medians
A
D
AD
A
D
and
B
E
BE
BE
, holds that
∠
C
A
D
=
∠
C
B
E
=
3
0
o
\angle CAD= \angle CBE=30^o
∠
C
A
D
=
∠
CBE
=
3
0
o
. Prove that triangle
A
B
C
ABC
A
BC
is equilateral.
7 divides (1^n+2^n+3^n)
For which
n
n
n
,
n
∈
N
n \in \mathbb{N}
n
∈
N
is the number
1
n
+
2
n
+
3
n
1^n+2^n+3^n
1
n
+
2
n
+
3
n
divisible by
7
7
7
?
y^2f(x)(f(x)-2x)\le (1-xy)(1+xy)
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
that satisfy
y
2
f
(
x
)
(
f
(
x
)
−
2
x
)
≤
(
1
−
x
y
)
(
1
+
x
y
)
y^2f(x)(f(x)-2x)\le (1-xy)(1+xy)
y
2
f
(
x
)
(
f
(
x
)
−
2
x
)
≤
(
1
−
x
y
)
(
1
+
x
y
)
for any
x
,
y
∈
R
x,y \in\mathbb{R}
x
,
y
∈
R
.
2
3
Hide problems
a/(b^3-1)+b/(a^3-1)=2(ab-2)/ (a^2b^2+3) 1990 Greece MO Grade X p2
If
a
+
b
=
1
a+b=1
a
+
b
=
1
,
∈
R
\in \mathbb{R}
∈
R
and
a
b
≠
0
ab \ne 0
ab
=
0
, prove that
a
b
3
−
1
+
b
a
3
−
1
=
2
(
a
b
−
2
)
a
2
b
2
+
3
\frac{a}{b^3-1}+\frac{b}{a^3-1}=\frac{2(ab-2)}{a^2b^2+3}
b
3
−
1
a
+
a
3
−
1
b
=
a
2
b
2
+
3
2
(
ab
−
2
)
\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}
Find all real solutions of
x
−
1
+
x
2
−
1
=
x
3
\sqrt{x-1}+\sqrt{x^2-1}=\sqrt{x^3}
x
−
1
+
x
2
−
1
=
x
3
vectod OK x OL+OL x OM + OM x On + ON x OK = fixed, midpoints of square
Let
A
C
B
D
ACBD
A
CB
D
be a asquare and
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
be points of
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
respectively. If
O
O
O
is the center of the square , prove that the expression
O
K
→
⋅
O
L
→
+
O
L
→
⋅
O
M
→
+
O
M
→
⋅
O
N
→
+
O
N
→
⋅
O
K
→
\overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}
O
K
⋅
O
L
+
O
L
⋅
OM
+
OM
⋅
ON
+
ON
⋅
O
K
is independent of positions of
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
, (i.e. is constant )
1
3
Hide problems
a=b if |a-b|<x for any x>0 1990 Greece MO Grade X p1
Let
a
,
b
a,b
a
,
b
, be two real numbers. If for any
x
>
0
x>0
x
>
0
holds that
∣
a
−
b
∣
<
x
|a-b|<x
∣
a
−
b
∣
<
x
, then prove that
a
=
b
a=b
a
=
b
.
BD/CD<HD/MD in right triangle, altitude, bisector, median from A
Let
A
B
C
ABC
A
BC
be a right triangle with
∠
A
=
9
0
o
\angle A=90^o
∠
A
=
9
0
o
and
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
A
H
,
A
D
,
A
M
AH,AD,AM
A
H
,
A
D
,
A
M
be altitude, angle bisector and median respectively. Prove that
B
D
C
D
<
H
D
M
D
.
\frac{BD}{CD}<\frac{HD}{MD}.
C
D
B
D
<
M
D
HD
.
A^2=0 if A^3=0 for 2x2 real matric A
Let
A
A
A
be a
2
x
2
2\,x\,2
2
x
2
matrix with real numbers. Prove that if
A
3
=
O
A^3=\mathbb{O}
A
3
=
O
then
A
2
=
O
A^2=\mathbb{O}
A
2
=
O
.
4
3
Hide problems
f(x+y)=f(x^2)+f(y^2)
Find all functions
f
:
R
+
→
R
f: \mathbb{R}^+\to\mathbb{R}
f
:
R
+
→
R
such that
f
(
x
+
y
)
=
f
(
x
2
)
+
f
(
y
2
)
f(x+y)=f(x^2)+f(y^2)
f
(
x
+
y
)
=
f
(
x
2
)
+
f
(
y
2
)
for any
x
,
y
∈
R
+
x,y \in\mathbb{R}^+
x
,
y
∈
R
+
last two digits of 6^{1989} - 1990 Greece MO Grade X p4
Since this is the
6
6
6
th Greek Math Olympiad and the year is
1989
1989
1989
, can you find the last two digits of
6
1989
6^{1989}
6
1989
?
{x} +{1/x} =1 for rationals
Froa nay real
x
x
x
, we denote
[
x
]
[x]
[
x
]
, the integer part of
x
x
x
and with
{
x
}
\{x\}
{
x
}
the fractional part of
x
x
x
, such that
x
=
[
x
]
+
{
x
}
x=[x]+\{x\}
x
=
[
x
]
+
{
x
}
. a) Find at least one real
x
x
x
such that
{
x
}
+
{
1
x
}
=
1
\{x\}+\left\{\frac{1}{x}\right\}=1
{
x
}
+
{
x
1
}
=
1
b) Find all rationals
x
x
x
such that
{
x
}
+
{
1
x
}
=
1
\{x\}+\left\{\frac{1}{x}\right\}=1
{
x
}
+
{
x
1
}
=
1