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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1987 Greece National Olympiad
1987 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
3
Hide problems
no of diagonals, in polygons by a diagonal in 100-gon 1987 Greece MO Grade X p4
Consider a convex
100
100
100
-gon
A
1
A
2
.
.
.
A
100
A_1A_2...A_{100}
A
1
A
2
...
A
100
. Draw the diagonal
A
43
A
81
A_{43}A_{81}
A
43
A
81
which divides it into two convex polygons
P
1
,
P
2
P_1,P_2
P
1
,
P
2
. How many vertices and how diagonals, has each of the polygons
P
1
,
P
2
P_1,P_2
P
1
,
P
2
?
MA/ AA_1+ MB/ BB_1> 2 MG/ GG_1
Let
A
,
B
A,B
A
,
B
be two points interior of circle
C
(
O
,
R
)
C(O,R)
C
(
O
,
R
)
and
M
M
M
a point on the circle. Let
A
1
,
B
1
A_1,B_1
A
1
,
B
1
be the intersections of the circle with lines
M
A
MA
M
A
,
M
B
MB
MB
respectively. Let
G
G
G
be the midpoint of
A
B
AB
A
B
and
G
1
=
C
∩
M
G
G_1= C\cap MG
G
1
=
C
∩
MG
. Prove that
M
A
A
A
1
+
M
B
B
B
1
>
2
M
G
G
G
1
\frac{MA}{AA_1}+ \frac{MB}{BB_1}> 2\frac{MG}{GG_1}
A
A
1
M
A
+
B
B
1
MB
>
2
G
G
1
MG
triangle with min area, Oxy
In rectangular coodinate system
O
x
y
Oxy
O
x
y
, consider the line
y
=
3
x
y=3x
y
=
3
x
and point
A
(
4
,
3
)
A(4,3)
A
(
4
,
3
)
. Find on the line
y
=
3
x
y=3x
y
=
3
x
, point
B
≠
O
B\ne O
B
=
O
such that the area of triangle
O
B
C
OBC
OBC
is the minimum possible, where
C
=
A
B
∩
O
x
C= AB\cap Ox
C
=
A
B
∩
O
x
.
3
3
Hide problems
\sqrt{a+x}+ \sqrt{a-x}>a
Solve for real values of parameter
a
a
a
, the inequality :
a
+
x
+
a
−
x
>
a
,
x
∈
R
\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}
a
+
x
+
a
−
x
>
a
,
x
∈
R
min (a-1)(a-3)(a-4)(a-6)+10 1987 Greece MO Grade X p3
Prova that for any real
a
a
a
, expresssion
A
=
(
a
−
1
)
(
a
−
3
)
(
a
−
4
)
(
a
−
6
)
+
10
A=(a-1)(a-3)(a-4)(a-6)+10
A
=
(
a
−
1
)
(
a
−
3
)
(
a
−
4
)
(
a
−
6
)
+
10
is always positive. What is the minimum value that expression
A
A
A
can take and for which values of
a
a
a
?
x_n+x_{k}=x_{nk}, strictly increasing with terms naturals
There is no sequence
x
n
x_n
x
n
strictly increasing with terms natural numbers such that :
x
n
+
x
k
=
x
n
k
,
f
o
r
a
n
y
n
,
k
∈
N
∗
x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*
x
n
+
x
k
=
x
nk
,
f
or
an
y
n
,
k
∈
N
∗
2
3
Hide problems
25/2 (n+2-\sqrt{2n+3})=q^2 1987 Greece MO Grade X p2
Prove that exprssion
A
=
25
2
(
n
+
2
−
2
n
+
3
)
A=\frac{25}{2}(n+2-\sqrt{2n+3})
A
=
2
25
(
n
+
2
−
2
n
+
3
)
,
(
n
∈
N
)
(n\in\mathbb{N})
(
n
∈
N
)
is a perfect square of an integer if exprssion
A
A
A
is an integer .
f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0 1987 Greece MO Grade XI p2
If for function
f
f
f
holds that
f
(
x
)
+
f
(
x
+
1
)
+
f
(
x
+
2
)
+
.
.
.
+
f
(
x
+
1986
)
=
0
f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0
f
(
x
)
+
f
(
x
+
1
)
+
f
(
x
+
2
)
+
...
+
f
(
x
+
1986
)
=
0
for any
∈
R
\in\mathbb{R}
∈
R
, prove that
f
f
f
is periodic and find one period of her.
determinant AB=0
Let
A
=
(
α
i
j
)
A=(\alpha_{ij})
A
=
(
α
ij
)
be a
m
x
n
m\,x\,n
m
x
n
matric and
B
=
(
β
k
l
)
B=(\beta_{kl})
B
=
(
β
k
l
)
be a
n
x
m
n\,x\, m
n
x
m
matric with
m
>
n
m>n
m
>
n
. Prove that
D
(
A
⋅
B
)
=
0
D(A\cdot B)=0
D
(
A
⋅
B
)
=
0
.
1
3
Hide problems
max n, convex n-gon with equal diagonals 1987 Greece MO Grade X p1
It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural
n
n
n
such that a convex
n
n
n
-gon has all it's diagonals equal.
color plane twith 3 colours 1987 Greece MO Grade XI p1
We color all points of a plane using
3
3
3
colors. Prove that there are at least two points of the plane having same colours and with distance among them
1
1
1
.
gcd = sum =\lambda_i a_i
a) Prove that every sub-group
(
A
,
+
)
(A,+)
(
A
,
+
)
of group
(
Z
,
+
)
(\mathbb{Z},+)
(
Z
,
+
)
is in the form
A
=
n
⋅
Z
A=n \cdot \mathbb{Z}
A
=
n
⋅
Z
for some
n
∈
Z
n \in \mathbb{Z}
n
∈
Z
where
n
⋅
Z
=
{
n
⋅
x
/
x
∈
Z
}
n \cdot \mathbb{Z}=\{n \cdot x/x\in\mathbb{Z}\}
n
⋅
Z
=
{
n
⋅
x
/
x
∈
Z
}
.b) Using problem (a) , prove that the greatest common divisor
d
d
d
of non zero integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,... ,a_n
a
1
,
a
2
,
...
,
a
n
is given by relation
d
=
λ
1
a
1
+
λ
2
a
2
+
.
.
.
λ
n
a
n
d=\lambda_1a_1+\lambda_2 a_2+...\lambda_n a_n
d
=
λ
1
a
1
+
λ
2
a
2
+
...
λ
n
a
n
with
λ
i
∈
Z
,
i
=
1
,
2
,
.
.
.
,
n
\lambda_i\in\mathbb{Z}, \,\, i=1,2,...,n
λ
i
∈
Z
,
i
=
1
,
2
,
...
,
n