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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1988 Greece National Olympiad
1988 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
3
Hide problems
k^2+2m, m^2+2k never both perfect squares 1988 Greece MO Grade XI p4
Prove that there are do not exist natural numbers
k
,
m
k, m
k
,
m
such that numbers
k
2
+
2
m
k^2+2m
k
2
+
2
m
,
m
2
+
2
k
m^2+2k
m
2
+
2
k
to be squares of integers.
\sqrt[ 2^{1453}]{2^{1821}} \in A, if 1,2 \in A and \sqrt{ab} \in A for a,b \in A
Let
A
⊆
R
A\subseteq \mathbb{R}
A
⊆
R
such that:i) If
a
,
b
∈
A
a,b\in A
a
,
b
∈
A
then
a
b
∈
A
\sqrt{ab} \in A
ab
∈
A
ii)
1
∈
A
1\in A
1
∈
A
and
2
∈
A
2\in A
2
∈
A
Prove that
2
1821
2
1453
∈
A
\sqrt[\displaystyle 2^{1453}]{2^{1821}}\in A
2
1453
2
1821
∈
A
.
lim sum 1/ (a_i ... a_n) when a_{n+1}= a^2_{n}-2
Let
a
1
=
5
a_1=5
a
1
=
5
and
a
n
+
1
=
a
n
2
−
2
a_{n+1}= a^2_{n}-2
a
n
+
1
=
a
n
2
−
2
for any
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
.a) Find
lim
n
→
∞
a
n
+
1
a
1
a
2
.
.
.
a
n
\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}
lim
n
→
∞
a
1
a
2
...
a
n
a
n
+
1
b) Find
lim
ν
→
∞
(
1
a
1
+
1
a
1
a
2
+
.
.
.
+
1
a
1
a
2
.
.
.
a
ν
)
\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)
lim
ν
→
∞
(
a
1
1
+
a
1
a
2
1
+
...
+
a
1
a
2
...
a
ν
1
)
3
3
Hide problems
min max distance 2 circles 1988 Greece MO Grade X p3
Two circles
(
O
1
,
R
1
)
(O_1,R_1)
(
O
1
,
R
1
)
,
(
O
2
,
R
2
)
(O_2,R_2)
(
O
2
,
R
2
)
lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles
MB/ MC+ ND/NC >1, rectangle ABCD
Bisectors of
∠
B
A
C
\angle BAC
∠
B
A
C
,
∠
C
A
D
\angle CAD
∠
C
A
D
in a rectangle
A
B
C
D
ABCD
A
BC
D
, intersect the sides
B
C
BC
BC
,
C
D
CD
C
D
at points
M
M
M
and
N
N
N
resp. Prove that
(
M
B
)
(
M
C
)
+
(
N
D
)
(
N
C
)
>
1
\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1
(
MC
)
(
MB
)
+
(
NC
)
(
N
D
)
>
1
matrix A^{3n} takes only two values if A^2+I=A
Let
A
A
A
be a
n
×
n
n \times n
n
×
n
matrix of real numbers such that
A
2
+
I
=
A
A^2+\mathbb{I}=A
A
2
+
I
=
A
, where
I
\mathbb{I}
I
is the identity
n
×
n
n \times n
n
×
n
matrix. Prove that the matrix
A
3
n
A^{3n}
A
3
n
, where
ν
∈
Z
\nu\in\mathbb{Z}
ν
∈
Z
takes only two values and find those values.
2
3
Hide problems
AD=AE if AB=AC, <BAD = 2<CDE 1988 Greece MO Grade XI p2
In isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
, consider point
D
D
D
on the base
B
C
BC
BC
and point
E
E
E
on side
A
C
AC
A
C
such that
∠
B
A
D
=
2
∠
C
D
E
\angle BAD = 2 \angle CDE
∠
B
A
D
=
2∠
C
D
E
. Prove that
A
D
=
A
E
AD=AE
A
D
=
A
E
.
BC^2/ MZ^2 >= 8 R U_a / (MD) (ME)
Let
A
B
C
ABC
A
BC
be a triangle inscribed in circle
C
(
O
,
R
)
C(O,R)
C
(
O
,
R
)
. Let
M
M
M
ba apoint on the arc
B
C
BC
BC
. Let
D
,
E
,
Z
D,E,Z
D
,
E
,
Z
be the feet of the perpendiculars drawn from
M
M
M
on lines
A
B
,
A
C
,
B
C
AB,AC,BC
A
B
,
A
C
,
BC
respectively. Prove that
(
B
C
)
2
(
M
Z
)
2
≥
8
R
U
a
(
M
D
)
⋅
(
M
E
)
\frac{(BC)^2}{(MZ)^2} \ge 8\frac{R U_a}{(MD)\cdot(ME)}
(
MZ
)
2
(
BC
)
2
≥
8
(
M
D
)
⋅
(
ME
)
R
U
a
where
U
a
U_a
U
a
is the altitude drawn on
B
C
BC
BC
.
vector locus |sum MA_i |<=1987, regular 1987-gon
Given regular
1987
1987
1987
-gon on plane with vertices
A
1
,
A
2
,
.
.
.
,
A
1987
A_1, A_2,..., A_{1987}
A
1
,
A
2
,
...
,
A
1987
. Find locus of points M of the plane sych that
∣
M
A
1
→
+
M
A
2
→
+
.
.
.
+
M
A
1987
→
∣
≤
1987
\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987
M
A
1
+
M
A
2
+
...
+
M
A
1987
≤
1987
.
1
3
Hide problems
\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c} 1988 Greece MO Grade X p1
Let
a
>
0
,
b
>
0
,
c
>
0
a>0,b>0,c>0
a
>
0
,
b
>
0
,
c
>
0
and
1987
+
a
+
1987
+
b
=
2
1987
+
c
\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}
1987
+
a
+
1987
+
b
=
2
1987
+
c
. Prove that
1
2
(
a
+
b
)
≥
c
\frac{1}{2} (a+b )\ge c
2
1
(
a
+
b
)
≥
c
.
min x^2+y^2, x+y=2a-4, xy=a^2-3a+5 - 1988 Greece MO Grade XI p1
Given
x
,
y
,
a
∈
R
x,y,a\in \mathbb{R}
x
,
y
,
a
∈
R
,
x
+
y
=
2
a
−
4
x+y=2a-4
x
+
y
=
2
a
−
4
and
x
y
=
a
2
−
3
a
+
5
xy=a^2-3a+5
x
y
=
a
2
−
3
a
+
5
. What is the minimum value of
x
2
+
y
2
x^2+y^2
x
2
+
y
2
?
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
that satidfy :
2
f
(
x
+
y
+
x
y
)
=
a
f
(
x
)
+
b
f
(
y
)
+
f
(
x
y
)
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)
2
f
(
x
+
y
+
x
y
)
=
a
f
(
x
)
+
b
f
(
y
)
+
f
(
x
y
)
for any
x
,
y
∈
R
x,y \in\mathbb{R}
x
,
y
∈
R
όπου
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
with
a
2
−
a
≠
b
2
−
b
a^2-a\ne b^2-b
a
2
−
a
=
b
2
−
b