MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2009 Greece Junior Math Olympiad
2009 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
Hide problems
k pupils move on a 3x3 grid only up and right , at least 2 same path
In the figure we see the paths connecting the square of a city (point
P
P
P
) with the school (point
S
S
S
). In the square there are
k
k
k
pupils starting to go to the school. They have the ability to move only to the right and up. If the pupils are free to choose any allowed path (in order to get to school), determine the minimum value of
k
k
k
so that in any case at least two pupils follow the same path. https://cdn.artofproblemsolving.com/attachments/e/2/b5d6c6db5942cb706428cb63af3ca15590727f.png
3
1
Hide problems
Algebra expressions
Consider the numbers
A
=
1
4
⋅
3
6
⋅
5
8
⋅
.
.
.
595
598
⋅
597
600
A= \frac{1}{4}\cdot \frac{3}{6}\cdot \frac{5}{8}\cdot ...\frac{595}{598}\cdot \frac{597}{600}
A
=
4
1
⋅
6
3
⋅
8
5
⋅
...
598
595
⋅
600
597
and
B
=
2
5
⋅
4
7
⋅
6
9
⋅
.
.
.
596
599
⋅
598
601
B= \frac{2}{5}\cdot \frac{4}{7}\cdot \frac{6}{9}\cdot ...\frac{596}{599}\cdot \frac{598}{601}
B
=
5
2
⋅
7
4
⋅
9
6
⋅
...
599
596
⋅
601
598
. Prove that: (a)
A
<
B
A < B
A
<
B
, (b)
A
<
1
5990
A < \frac{1}{5990}
A
<
5990
1
1
1
Hide problems
Easy Number Theory
If the number
K
=
9
n
2
+
31
n
2
+
7
K = \frac{9n^2+31}{n^2+7}
K
=
n
2
+
7
9
n
2
+
31
is integer, find the possible values of
n
∈
Z
n \in Z
n
∈
Z
.
2
1
Hide problems
angle chasing inside an equilateral, BD+DC=BC, BA=BE (Greece Junior 2009)
From vertex
A
A
A
of an equilateral triangle
A
B
C
ABC
A
BC
, a ray
A
x
Ax
A
x
intersects
B
C
BC
BC
at point
D
D
D
. Let
E
E
E
be a point on
A
x
Ax
A
x
such that
B
A
=
B
E
BA =BE
B
A
=
BE
. Calculate
∠
A
E
C
\angle AEC
∠
A
EC
.