MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2014 Greece National Olympiad
2014 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Problem 4
We are given a circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
and two points
A
,
B
A,B
A
,
B
so that
R
<
A
B
<
2
R
R<AB<2R
R
<
A
B
<
2
R
.The circle
c
1
(
A
,
r
)
c_1 (A,r)
c
1
(
A
,
r
)
(
0
<
r
<
R
0<r<R
0
<
r
<
R
) crosses the circle
c
c
c
at C,D (
C
C
C
belongs to the short arc
A
B
AB
A
B
).From
B
B
B
we consider the tangent lines
B
E
,
B
F
BE,BF
BE
,
BF
to the circle
c
1
c_1
c
1
,in such way that
E
E
E
lays out of the circle
c
c
c
.If
M
≡
E
C
∩
D
F
M\equiv EC\cap DF
M
≡
EC
∩
D
F
show that the quadrilateral
B
C
F
M
BCFM
BCFM
is cyclic.
3
1
Hide problems
Sum in black and white squares
For even positive integer
n
n
n
we put all numbers
1
,
2
,
.
.
.
,
n
2
1,2,...,n^2
1
,
2
,
...
,
n
2
into the squares of an
n
×
n
n\times n
n
×
n
chessboard (each number appears once and only once). Let
S
1
S_1
S
1
be the sum of the numbers put in the black squares and
S
2
S_2
S
2
be the sum of the numbers put in the white squares. Find all
n
n
n
such that we can achieve
S
1
S
2
=
39
64
.
\frac{S_1}{S_2}=\frac{39}{64}.
S
2
S
1
=
64
39
.
2
1
Hide problems
"Cubic" fraction
Find all the integers
n
n
n
for which
8
n
−
25
n
+
5
\frac{8n-25}{n+5}
n
+
5
8
n
−
25
is cube of a rational number.
1
1
Hide problems
Polynomials
Find all the polynomials with real coefficients which satisfy
(
x
2
−
6
x
+
8
)
P
(
x
)
=
(
x
2
+
2
x
)
P
(
x
−
2
)
(x^2-6x+8)P(x)=(x^2+2x)P(x-2)
(
x
2
−
6
x
+
8
)
P
(
x
)
=
(
x
2
+
2
x
)
P
(
x
−
2
)
for all
x
∈
R
x\in \mathbb{R}
x
∈
R
.