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Romania Contests
Romania National Olympiad
1997 Romania National Olympiad
1997 Romania National Olympiad
Part of
Romania National Olympiad
Subcontests
(4)
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4
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altitudes wanted in triangle - easy roman geo - 1997 Romania NMO VII p3
The triangle
A
B
C
ABC
A
BC
has
∠
A
C
B
=
3
0
o
\angle ACB = 30^o
∠
A
CB
=
3
0
o
,
B
C
=
4
BC = 4
BC
=
4
cm and
A
B
=
3
AB = 3
A
B
=
3
cm . Compute the altitudes of the triangle.
parallelepiped with min CF + FM - 1997 Romania NMO VIII p3
A
B
C
D
A
′
B
′
C
D
′
ABCDA'B'CD'
A
BC
D
A
′
B
′
C
D
′
is a rectangular parallelepiped with
A
A
′
=
2
A
B
=
8
a
AA'= 2AB = 8a
A
A
′
=
2
A
B
=
8
a
,
E
E
E
is the midpoint of
(
A
B
)
(AB)
(
A
B
)
and
M
M
M
is the point of
(
D
D
′
)
(DD')
(
D
D
′
)
for which
D
M
=
a
(
1
+
A
D
A
C
)
DM = a \left( 1 + \frac{AD}{AC}\right)
D
M
=
a
(
1
+
A
C
A
D
)
.a) Find the position of the point.
F
F
F
on the segment
(
A
A
′
)
(AA')
(
A
A
′
)
for which the sum
C
F
+
F
M
CF + FM
CF
+
FM
has the minimum possible value.b) Taking
F
F
F
as above, compute the measure of the angle of the planes
(
D
,
E
,
F
)
(D, E, F)
(
D
,
E
,
F
)
and
(
D
,
B
′
,
C
′
)
(D, B', C')
(
D
,
B
′
,
C
′
)
.c) Knowing that the straight lines
A
C
′
AC'
A
C
′
and
F
D
FD
F
D
are perpendicular, compute the volume of the parallelepiped
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
.
f(u)+f(v)=7
Suppose that
a
,
b
,
c
,
d
∈
R
a,b,c,d\in\mathbb{R}
a
,
b
,
c
,
d
∈
R
and
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x)=ax^3+bx^2+cx+d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
such that
f
(
2
)
+
f
(
5
)
<
7
<
f
(
3
)
+
f
(
4
)
f(2)+f(5)<7<f(3)+f(4)
f
(
2
)
+
f
(
5
)
<
7
<
f
(
3
)
+
f
(
4
)
. Prove that there exists
u
,
v
∈
R
u,v\in\mathbb{R}
u
,
v
∈
R
such that
u
+
v
=
7
,
f
(
u
)
+
f
(
v
)
=
7
u+v=7 , f(u)+f(v)=7
u
+
v
=
7
,
f
(
u
)
+
f
(
v
)
=
7