MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2013 Greece National Olympiad
2013 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
1
Hide problems
Minimum number of sets!
We define the sets
A
1
,
A
2
,
.
.
.
,
A
160
A_1,A_2,...,A_{160}
A
1
,
A
2
,
...
,
A
160
such that
∣
A
i
∣
=
i
\left|A_{i} \right|=i
∣
A
i
∣
=
i
for all
i
=
1
,
2
,
.
.
.
,
160
i=1,2,...,160
i
=
1
,
2
,
...
,
160
. With the elements of these sets we create new sets
M
1
,
M
2
,
.
.
.
M
n
M_1,M_2,...M_n
M
1
,
M
2
,
...
M
n
by the following procedure: in the first step we choose some of the sets
A
1
,
A
2
,
.
.
.
,
A
160
A_1,A_2,...,A_{160}
A
1
,
A
2
,
...
,
A
160
and we remove from each of them the same number of elements. These elements that we removed are the elements of
M
1
M_1
M
1
. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define
M
2
M_2
M
2
. We continue similarly until there are no more elements in
A
1
,
A
2
,
.
.
.
,
A
160
A_1,A_2,...,A_{160}
A
1
,
A
2
,
...
,
A
160
, thus defining the sets
M
1
,
M
2
,
.
.
.
,
M
n
M_1,M_2,...,M_n
M
1
,
M
2
,
...
,
M
n
. Find the minimum value of
n
n
n
.
4
1
Hide problems
Equal triangles
Let a triangle
A
B
C
ABC
A
BC
inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
and
D
D
D
an arbitrary point on
B
C
BC
BC
(different from the midpoint).The circumscribed circle of
B
O
D
BOD
BO
D
,which is
(
c
1
)
(c_1)
(
c
1
)
, meets
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at
K
K
K
and
A
B
AB
A
B
at
Z
Z
Z
.The circumscribed circle of
C
O
D
COD
CO
D
(
c
2
)
(c_2)
(
c
2
)
,meets
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at
M
M
M
and
A
C
AC
A
C
at
E
E
E
.Finally, the circumscribed circle of
A
E
Z
AEZ
A
EZ
(
c
3
)
(c_3)
(
c
3
)
,meets
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at
N
N
N
.Prove that
△
A
B
C
=
△
K
M
N
.
\triangle{ABC}=\triangle{KMN}.
△
A
BC
=
△
K
MN
.
2
1
Hide problems
Equation in Z!
Solve in integers the following equation:
y
=
2
x
2
+
5
x
y
+
3
y
2
y=2x^2+5xy+3y^2
y
=
2
x
2
+
5
x
y
+
3
y
2
1
1
Hide problems
Real sequence
Let the sequence of real numbers
(
a
n
)
,
n
=
1
,
2
,
3...
(a_n),n=1,2,3...
(
a
n
)
,
n
=
1
,
2
,
3...
with
a
1
=
2
a_1=2
a
1
=
2
and
a
n
=
(
n
+
1
n
−
1
)
(
a
1
+
a
2
+
.
.
.
+
a
n
−
1
)
,
n
≥
2
a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2
a
n
=
(
n
−
1
n
+
1
)
(
a
1
+
a
2
+
...
+
a
n
−
1
)
,
n
≥
2
. Find the term
a
2013
a_{2013}
a
2013
.