MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2023 Greece Junior Math Olympiad
2023 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
2
1
Hide problems
BD=2 EN if CD // ME , AM=//MB, AN=//NC
In triangle
A
B
C
ABC
A
BC
, points
M
M
M
,
N
N
N
are the midpoints of sides
A
B
AB
A
B
,
A
C
AC
A
C
respelctively. Let
D
D
D
and
E
E
E
be two points on line segment
B
N
BN
BN
such that
C
D
∥
M
E
CD \parallel ME
C
D
∥
ME
and
B
D
<
B
E
BD <BE
B
D
<
BE
. Prove that
B
D
=
2
⋅
E
N
BD=2\cdot EN
B
D
=
2
⋅
EN
.
3
1
Hide problems
number of rectangles by lattice vertices
Find the number of rectangles who have the following properties: a) Have for vertices, points
(
x
,
y
)
(x,y)
(
x
,
y
)
of plane
O
x
y
Oxy
O
x
y
with
x
,
y
x,y
x
,
y
non negative integers and
x
≤
8
x \le 8
x
≤
8
,
y
≤
8
y\le 8
y
≤
8
b) Have sides parallel to axes c) Have area
E
E
E
, with
30
<
E
≤
40
30<E\le 40
30
<
E
≤
40
4
1
Hide problems
b | (a-1), (2a+1) | (5b-3), a>1
Find all positive integers
a
,
b
a,b
a
,
b
with
a
>
1
a>1
a
>
1
such that,
b
b
b
is a divisor of
a
−
1
a-1
a
−
1
and
2
a
+
1
2a+1
2
a
+
1
is a divisor of
5
b
−
3
5b-3
5
b
−
3
.
1
1
Hide problems
a+b+c=0, ab^3+bc^3+ca^3=0 , system
Solve in real numbers the system:
{
a
+
b
+
c
=
0
a
b
3
+
b
c
3
+
c
a
3
=
0
\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}
{
a
+
b
+
c
=
0
a
b
3
+
b
c
3
+
c
a
3
=
0