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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1989 Greece National Olympiad
1989 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
3
Hide problems
CD >DE if 2AD> AC+ AE
From a point
A
A
A
not on line
ε
\varepsilon
ε
, we drop the perpendicular
A
B
AB
A
B
on
ε
\varepsilon
ε
and three other not perpendicular lines
A
C
AC
A
C
,
A
D
AD
A
D
,
A
E
AE
A
E
which lie on the same semiplane defines by
A
B
AB
A
B
, such that
(
A
D
)
>
1
2
(
(
A
C
)
+
(
A
E
)
)
(AD )>\frac{1}{2}((AC)+(AE))
(
A
D
)
>
2
1
((
A
C
)
+
(
A
E
))
. Prove that
(
C
D
)
>
(
D
E
)
.
(CD )>(DE).
(
C
D
)
>
(
D
E
)
.
(Points
B
,
C
,
D
,
,
E
B,C,D,,E
B
,
C
,
D
,,
E
lie on line
ε
\varepsilon
ε
) .
a^4+ a^3-10 a^2+9 a+4>0 for a>=0
If
a
≥
0
a\ge 0
a
≥
0
prove that
a
4
+
a
3
−
10
a
2
+
9
a
+
4
>
0
a^4+ a^3-10 a^2+9 a+4>0
a
4
+
a
3
−
10
a
2
+
9
a
+
4
>
0
.
x_{n+2}=1/12 x_{n+1}+1/2 x_{n}+1, limit
Find the limit of the sequence
x
n
x_n
x
n
defined by recurrence relation
x
n
+
2
=
1
12
x
n
+
1
+
1
2
x
n
+
1
x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1
x
n
+
2
=
12
1
x
n
+
1
+
2
1
x
n
+
1
where
n
=
0
,
1
,
2
,
.
.
.
n=0,1,2,...
n
=
0
,
1
,
2
,
...
for any initial values
x
2
,
x
1
x_2,x_1
x
2
,
x
1
.
4
3
Hide problems
2|3-2x|-|x-2|=x 1989 Greece MO Grade X p4
Solve
2
∣
3
−
2
x
∣
−
∣
x
−
2
∣
=
x
2|3-2x|-|x-2|=x
2∣3
−
2
x
∣
−
∣
x
−
2∣
=
x
,
x
∈
R
x\in\mathbb{R}
x
∈
R
.
h^2 <= ab for tangential trapezoid
A trapezoid with bases
a
,
b
a,b
a
,
b
and altitude
h
h
h
is circumscribed around a circl.. Prove that
h
2
≤
a
b
h^2\le ab
h
2
≤
ab
.
(x^ky)^2=e if x^{n}=e,y^2=e,yxy=x^{-1}
In a group
G
G
G
, we have two elements
x
,
y
x,y
x
,
y
such that
x
n
=
e
,
y
2
=
e
,
y
x
y
=
x
−
1
x^{n}=e,y^2=e,yxy=x^{-1}
x
n
=
e
,
y
2
=
e
,
y
x
y
=
x
−
1
,
n
≥
1
n\ge 1
n
≥
1
. Prove that for any
k
∈
N
k\in\mathbb{N}
k
∈
N
holds
(
x
k
y
)
2
=
e
(x^ky)^2=e
(
x
k
y
)
2
=
e
.Note : e=group's identity .
2
3
Hide problems
max no of stories starting at odd no 1989 Greece MO Grade X p2
A collection of short stories written by Papadiamantis contains
70
70
70
short stories, one of
1
1
1
page, one of
2
2
2
pages, ... one of
70
70
70
pages . and not nessecarily in that order. Every short story starts on a new page and numbering of pages of the book starts from the first page . What is the maximum number of short stories that start on page with odd number?
locus,(AM)^2+(AN)^2=2(AB)^2, AB=AC
Let
M
M
M
be a point on side
B
C
BC
BC
of isosceles
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB=AC
A
B
=
A
C
) and let
N
N
N
be a points on the extension of
B
C
BC
BC
such that
(
A
M
)
2
+
(
A
N
)
2
=
2
(
A
B
)
2
(AM)^2+(AN)^2=2(AB)^2
(
A
M
)
2
+
(
A
N
)
2
=
2
(
A
B
)
2
. Find the locus of point
N
N
N
when point
M
M
M
moves on side
B
C
BC
BC
.
centroids lattice points, 70 lattice points
On the plane we consider
70
70
70
points
A
1
,
A
2
,
.
.
.
,
A
70
A_1,A_2,...,A_{70}
A
1
,
A
2
,
...
,
A
70
with integer coodinates. Suppose each pooints has weight
1
1
1
and the centers of gravity of the triangles
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
,
A
2
A
3
A
4
A_2A_3A_4
A
2
A
3
A
4
,
.
.
..
..
.,
A
68
A
69
A
70
A_{68}A_{69}A_{70}
A
68
A
69
A
70
,
A
69
A
70
A
1
A_{69}A_{70}A_{1}
A
69
A
70
A
1
,
A
70
A
1
A
2
A_{70}A_{1}A_{2}
A
70
A
1
A
2
have integer coodinates. Prove that the centers of gravity of any triple
A
i
,
A
j
,
.
.
.
,
A
k
A_i,A_j,...,A_{k}
A
i
,
A
j
,
...
,
A
k
has integer coodinates.
1
3
Hide problems
periodic wanted, g(x)=f(x+a), f even, g odd - 1989 Greece MO Grade XI p1
Consider two functions
f
,
g
:
R
→
R
f , \,g \,:\mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
such that from some
a
>
0
a>0
a
>
0
holds
g
(
x
)
=
f
(
x
+
a
)
g(x)=f(x+a)
g
(
x
)
=
f
(
x
+
a
)
for any
x
∈
R
x \in \mathbb{R}
x
∈
R
. If
f
f
f
is even and
g
g
g
is odd, prove that both functions are periodic.
4x8 system of equations 1989 Greece MO Grade X p1
Let
a
,
b
,
c
,
d
x
,
y
,
z
,
w
a,b,c,d x,y,z, w
a
,
b
,
c
,
d
x
,
y
,
z
,
w
be real numbers such that
a
x
−
b
y
−
c
z
−
d
w
=
0
b
x
+
a
y
−
d
z
+
c
w
=
0
c
x
+
d
y
+
a
z
−
b
w
=
0
d
x
−
c
y
+
b
z
+
a
w
=
0
\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}
a
x
−
b
y
−
cz
−
d
w
=
0
b
x
+
a
y
−
d
z
+
c
w
=
0
c
x
+
d
y
+
a
z
−
b
w
=
0
d
x
−
cy
+
b
z
+
a
w
=
0
prove that
a
=
b
=
c
=
d
=
0
,
o
r
x
=
y
=
z
=
w
=
0
a=b=c=d=0, \ \ or \ \ x=y=z=w=0
a
=
b
=
c
=
d
=
0
,
or
x
=
y
=
z
=
w
=
0
\sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}
Find all real solutions of
9
+
x
1
+
9
+
x
2
+
.
.
.
+
9
+
x
100
=
100
10
16
−
x
1
+
16
−
x
2
+
.
.
.
+
16
−
x
100
=
100
15
\begin{matrix} \sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\ \sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15} \end{matrix}
9
+
x
1
+
9
+
x
2
+
...
+
9
+
x
100
=
100
10
16
−
x
1
+
16
−
x
2
+
...
+
16
−
x
100
=
100
15