MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1995 Vietnam National Olympiad
1995 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
2
2
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Sequence of numbers
The sequence (a_n) is defined as follows: a_0\equal{}1, a_1\equal{}3 For
n
≥
2
n\ge 2
n
≥
2
, a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}9a_n if n is even, a_{n\plus{}2}\equal{}9a_{n\plus{}1}\plus{}5a_n if n is odd. Prove that 1) (a_{1995})^2\plus{}(a_{1996})^2\plus{}...\plus{}(a_{2000})^2 is divisible by 20 2) a_{2n\plus{}1} is not a perfect square for every natural numbers
n
n
n
.
greater than 1995
Find all poltnomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients satisfying: For all
a
>
1995
a>1995
a
>
1995
, the number of real roots of P(x)\equal{}a (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.
1
2
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Equation
Find all real solutions to x^3 \minus{} 3x^2 \minus{} 8x \plus{} 40 \minus{} 8\sqrt[4]{4x \plus{} 4} \equal{} 0
concurrence of four planes
Let a tetrahedron
A
B
C
D
ABCD
A
BC
D
and
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
be the circumcenters of triangles
B
C
D
,
C
D
A
,
D
A
B
,
A
B
C
BCD,CDA,DAB,ABC
BC
D
,
C
D
A
,
D
A
B
,
A
BC
respectively. Denote planes
(
P
A
)
,
(
P
B
)
,
(
P
C
)
,
(
P
D
)
(P_A),(P_B),(P_C),(P_D)
(
P
A
)
,
(
P
B
)
,
(
P
C
)
,
(
P
D
)
be the planes which pass through
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
and perpendicular to
C
′
D
′
,
D
′
A
′
,
A
′
B
′
,
B
′
C
′
C'D',D'A',A'B',B'C'
C
′
D
′
,
D
′
A
′
,
A
′
B
′
,
B
′
C
′
respectively. Prove that these planes have a common point called
I
.
I.
I
.
If
P
P
P
is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
3
2
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similar triangles
Let a non-equilateral triangle
A
B
C
ABC
A
BC
and
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
are its altitudes. On the rays
A
D
,
B
E
,
C
F
,
AD,BE,CF,
A
D
,
BE
,
CF
,
respectively, let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
such that \frac {AA'}{AD} \equal{} \frac {BB'}{BE} \equal{} \frac {CC'}{CF} \equal{} k. Find all values of
k
k
k
such that
△
A
′
B
′
C
′
∼
△
A
B
C
\triangle A'B'C'\sim\triangle ABC
△
A
′
B
′
C
′
∼
△
A
BC
for any non-triangle
A
B
C
.
ABC.
A
BC
.
Coloring regular 2n-polygon
Given an integer
n
≥
2
n\ge 2
n
≥
2
and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: (i) Each vertice is colored by exactly one color. (ii) Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. Alternative statement: In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.