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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1994 Vietnam National Olympiad
1994 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
x_{n+1}=f(x_n)
Define the sequence
{
x
n
}
\{x_{n}\}
{
x
n
}
by
x
0
=
a
∈
(
0
,
1
)
x_{0}=a\in (0,1)
x
0
=
a
∈
(
0
,
1
)
and
x
n
+
1
=
4
π
2
(
cos
−
1
x
n
+
π
2
)
sin
−
1
x
n
(
n
=
0
,
1
,
2
,
.
.
.
)
x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)
x
n
+
1
=
π
2
4
(
cos
−
1
x
n
+
2
π
)
sin
−
1
x
n
(
n
=
0
,
1
,
2
,
...
)
. Show that the sequence converges and find its limit.
Do there exist polynomials ?
Do there exist polynomials
p
(
x
)
,
q
(
x
)
,
r
(
x
)
p(x), q(x), r(x)
p
(
x
)
,
q
(
x
)
,
r
(
x
)
whose coefficients are positive integers such that
p
(
x
)
=
(
x
2
−
3
x
+
3
)
q
(
x
)
p(x) = (x^{2}-3x+3) q(x)
p
(
x
)
=
(
x
2
−
3
x
+
3
)
q
(
x
)
and
q
(
x
)
=
(
x
2
20
−
x
15
+
1
12
)
r
(
x
)
q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)
q
(
x
)
=
(
20
x
2
−
15
x
+
12
1
)
r
(
x
)
?
2
2
Hide problems
condition on $ABC$ for $A'B'C'$ to be equilateral.
A
B
C
ABC
A
BC
is a triangle. Reflect each vertex in the opposite side to get the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
. Find a necessary and sufficient condition on
A
B
C
ABC
A
BC
for
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
to be equilateral.
sphere
S
S
S
is a sphere center
O
.
G
O. G
O
.
G
and
G
′
G'
G
′
are two perpendicular great circles on
S
S
S
. Take
A
,
B
,
C
A, B, C
A
,
B
,
C
on
G
G
G
and
D
D
D
on
G
′
G'
G
′
such that the altitudes of the tetrahedron
A
B
C
D
ABCD
A
BC
D
intersect at a point. Find the locus of the intersection.
1
2
Hide problems
solve system
Find all real solutions to
x
3
+
3
x
−
3
+
ln
(
x
2
−
x
+
1
)
=
y
,
x^{3}+3x-3+\ln{(x^{2}-x+1)}=y,
x
3
+
3
x
−
3
+
ln
(
x
2
−
x
+
1
)
=
y
,
y
3
+
3
y
−
3
+
ln
(
y
2
−
y
+
1
)
=
z
,
y^{3}+3y-3+\ln{(y^{2}-y+1)}=z,
y
3
+
3
y
−
3
+
ln
(
y
2
−
y
+
1
)
=
z
,
z
3
+
3
z
−
3
+
ln
(
z
2
−
z
+
1
)
=
x
.
z^{3}+3z-3+\ln{(z^{2}-z+1)}=x.
z
3
+
3
z
−
3
+
ln
(
z
2
−
z
+
1
)
=
x
.
$n+1$ containers arranged in a circle
There are
n
+
1
n+1
n
+
1
containers arranged in a circle. One container has
n
n
n
stones, the others are empty. A move is to choose two containers
A
A
A
and
B
B
B
, take a stone from
A
A
A
and put it in one of the containers adjacent to
B
B
B
, and to take a stone from
B
B
B
and put it in one of the containers adjacent to
A
A
A
. We can take
A
=
B
A = B
A
=
B
. For which
n
n
n
is it possible by series of moves to end up with one stone in each container except that which originally held
n
n
n
stones.