MathDB
Problems
Contests
National and Regional Contests
Morocco Contests
Morocco National Olympiad
2005 Morocco National Olympiad
2005 Morocco National Olympiad
Part of
Morocco National Olympiad
Subcontests
(3)
1
1
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square and tangent line
In a square
A
B
C
D
ABCD
A
BC
D
let
F
F
F
be the midpoint of
[
C
D
]
\left[ CD\right]
[
C
D
]
and let
E
E
E
be a point on
[
A
B
]
\left[ AB\right]
[
A
B
]
such that
A
E
>
E
B
AE>EB
A
E
>
EB
. the parallel with
(
D
E
)
\left( DE\right)
(
D
E
)
passing by
F
F
F
meets the segment
[
B
C
]
\left[ BC\right]
[
BC
]
at
H
H
H
. Prove that the line
(
E
H
)
\left( EH\right)
(
E
H
)
is tangent to the circle circumscribed with
A
B
C
D
ABCD
A
BC
D
3
1
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Coloring Points
Consider
n
n
n
points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
on a circle. How many ways are there if we want to color these points by
p
p
p
colors, so that each two neighbors points are colored with two different colors?
2
1
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interesting...
Find all the positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfiing :
x
2
+
y
2
+
z
2
=
2
x
y
z
x^{2}+y^{2}+z^{2}=2xyz
x
2
+
y
2
+
z
2
=
2
x
yz