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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1991 Greece National Olympiad
1991 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
3
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3 white, 3 green, 3 red 1x1 in 3x3 - 1991 Greece MO Grade X p4
In how many ways can we construct a square with dimensions
3
×
3
3\times 3
3
×
3
using
3
3
3
white,
3
3
3
green and
3
3
3
red squares of dimensions
1
×
1
1\times 1
1
×
1
, such that in every horizontal and in every certical line, squares have different colours .
3^x+29=2^y
Find all positive intger solutions of
3
x
+
29
=
2
y
3^x+29=2^y
3
x
+
29
=
2
y
.
1^{1990}+2^{1990}+3^{1990}+...+1990^{1990} mod 10
If we divide number
1
1990
+
2
1990
+
3
1990
+
.
.
.
+
199
0
1990
1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}
1
1990
+
2
1990
+
3
1990
+
...
+
199
0
1990
with
10
10
10
, what remainder will we find?
3
3
Hide problems
overline{xxyy} - 1991 Greece MO Grade X p3
Find all 2-digit numbers
n
n
n
having the property:'Number
n
2
n^2
n
2
is 4-digit number of form
x
x
y
y
‾
\overline{xxyy}
xx
yy
.
exists triangle that can be partitions in 2050 congruent triangles.
Prove that exists triangle that can be partitions in
2050
2050
2050
congruent triangles.
4 white,4 green, 4 red and 4 blue 1x1 in 4x4
In how many ways can we construct a square with dimensions
4
×
4
4\times 4
4
×
4
using
4
4
4
white,
4
4
4
green ,
4
4
4
red and 4
b
l
u
e
blue
b
l
u
e
squares of dimensions
1
×
1
1\times 1
1
×
1
, such that in every horizontal and in every certical line, squares have different colours .
2
3
Hide problems
2 equidistant points wanted - 1991 Greece MO Grade X p2
Let
x
O
y
^
\widehat{xOy}
x
O
y
be an acute angle ,
A
A
A
a point on ray
O
y
Oy
O
y
and
B
B
B
a point on ray
O
x
Ox
O
x
such that
A
B
⊥
O
X
AB \perp OX
A
B
⊥
OX
.Prove that there are two points on
O
x
Ox
O
x
, each of the equidistant from
A
A
A
and
O
x
Ox
O
x
.
collinearity and concyclic wanted, 2 intersecting circles
Given two circles
(
C
1
)
(C_1)
(
C
1
)
and
(
C
2
)
(C_2)
(
C
2
)
with centers
O
1
\displaystyle{O_1}
O
1
and
O
2
O_2
O
2
respectively, intersecting at points
A
A
A
and
B
B
B
. Let
A
C
AC
A
C
και
A
D
AD
A
D
be the diameters of
(
C
1
)
(C_1)
(
C
1
)
and
(
C
2
)
(C_2)
(
C
2
)
respectively . Tangent line of circle
(
C
1
)
(C_1)
(
C
1
)
at point
A
A
A
intersects
(
C
2
)
(C_2)
(
C
2
)
at point
M
M
M
and tangent line of circle
(
C
2
)
(C_2)
(
C
2
)
at point A intersects
(
C
1
)
(C_1)
(
C
1
)
at point
N
N
N
. Let
P
P
P
be a point on line
A
B
AB
A
B
such that
A
B
=
B
P
AB=BP
A
B
=
BP
. Prove that: a) Points
B
,
C
,
D
B,C,D
B
,
C
,
D
are collinear. b) Quadrilateral
A
M
P
N
AMPN
A
MPN
is cyclic.
vertors OI = OA_1+OB_1 + OC_1, arc midpoints
Let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
and let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the midpoints of arcs
B
C
,
C
A
,
A
B
BC, CA,AB
BC
,
C
A
,
A
B
respectively. If
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
, prove that
O
I
→
=
O
A
1
→
+
O
B
1
→
+
O
C
1
→
.
\overrightarrow{OI}= \overrightarrow{OA_1}+ \overrightarrow{OB_1}+ \overrightarrow{OC_1}.
O
I
=
O
A
1
+
O
B
1
+
O
C
1
.
1
3
Hide problems
a+b>2 if a+b<2ab - 1991 Greece MO Grade X p1
Let
a
,
b
a, b
a
,
b
be two reals such that
a
+
b
<
2
a
b
a+b<2ab
a
+
b
<
2
ab
. Prove that
a
+
b
>
2
a+b>2
a
+
b
>
2
{f(f(x))=x+1, f: &Zeta;->&Zeta;
Prove that there is no function
f
:
Z
→
Z
f: \mathbb{Z}\to\mathbb{Z}
f
:
Z
→
Z
such that
f
(
f
(
x
)
)
=
x
+
1
f(f(x))=x+1
f
(
f
(
x
))
=
x
+
1
, for all
x
∈
Z
x\in\mathbb{Z}
x
∈
Z
.
P(x^3+1)=(P (x+1))^3
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
, such that
P
(
x
3
+
1
)
=
(
P
(
x
+
1
)
)
3
P(x^3+1)=\left(P (x+1)\right)^3
P
(
x
3
+
1
)
=
(
P
(
x
+
1
)
)
3