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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1995 Vietnam Team Selection Test
1995 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
(a^3 + b^3)^n = 4*(ab)^1995
Find all integers
a
a
a
,
b
b
b
,
n
n
n
greater than
1
1
1
which satisfy \left(a^3 \plus{} b^3\right)^n \equal{} 4(ab)^{1995}
n is periodic with the smallest period 1995
Consider the function f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}.
1.
1.
1.
Prove that there is a continuous function
g
(
x
)
g(x)
g
(
x
)
on
R
\mathbb{R}
R
satisfying f(g(x)) \equal{} x and
g
(
x
)
>
x
g(x) > x
g
(
x
)
>
x
for all real
x
x
x
.
2.
2.
2.
Show that there exists a real number
a
>
1
a > 1
a
>
1
such that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
, n \equal{} 1, 2, \ldots, defined as follows a_0 \equal{} a, a_{n \plus{} 1} \equal{} f(a_n),
∀
n
∈
N
\forall n\in\mathbb{N}
∀
n
∈
N
is periodic with the smallest period
1995
1995
1995
.
2
2
Hide problems
P(x) = x^{n + 1} + kx^n - 870x^2 + 1945x + 1995
Find all integers
k
k
k
such that for infinitely many integers
n
≥
3
n \ge 3
n
≥
3
the polynomial
P
(
x
)
=
x
n
+
1
+
k
x
n
−
870
x
2
+
1945
x
+
1995
P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995
P
(
x
)
=
x
n
+
1
+
k
x
n
−
870
x
2
+
1945
x
+
1995
can be reduced into two polynomials with integer coefficients.
Pair (n, p) of nonnegative integers is called nice
For any nonnegative integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the greatest integer such that 2^{f(n)} | n \plus{} 1. A pair
(
n
,
p
)
(n, p)
(
n
,
p
)
of nonnegative integers is called nice if
2
f
(
n
)
>
p
2^{f(n)} > p
2
f
(
n
)
>
p
. Find all triples
(
n
,
p
,
q
)
(n, p, q)
(
n
,
p
,
q
)
of nonnegative integers such that the pairs
(
n
,
p
)
(n, p)
(
n
,
p
)
,
(
p
,
q
)
(p, q)
(
p
,
q
)
and (n \plus{} p \plus{} q, n) are all nice.
1
2
Hide problems
Radical axes of the circumcircles of the pairs of triangles
Let be given a triangle
A
B
C
ABC
A
BC
with BC \equal{} a, CA \equal{} b, AB \equal{} c. Six distinct points
A
1
A_1
A
1
,
A
2
A_2
A
2
,
B
1
B_1
B
1
,
B
2
B_2
B
2
,
C
1
C_1
C
1
,
C
2
C_2
C
2
not coinciding with
A
A
A
,
B
B
B
,
C
C
C
are chosen so that
A
1
A_1
A
1
,
A
2
A_2
A
2
lie on line
B
C
BC
BC
;
B
1
B_1
B
1
,
B
2
B_2
B
2
lie on
C
A
CA
C
A
and
C
1
C_1
C
1
,
C
2
C_2
C
2
lie on
A
B
AB
A
B
. Let
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
three real numbers satisfy \overrightarrow{A_1A_2} \equal{} \frac {\alpha}{a}\overrightarrow{BC}, \overrightarrow{B_1B_2} \equal{} \frac {\beta}{b}\overrightarrow{CA}, \overrightarrow{C_1C_2} \equal{} \frac {\gamma}{c}\overrightarrow{AB}. Let
d
A
d_A
d
A
,
d
B
d_B
d
B
,
d
C
d_C
d
C
be respectively the radical axes of the circumcircles of the pairs of triangles
A
B
1
C
1
AB_1C_1
A
B
1
C
1
and
A
B
2
C
2
AB_2C_2
A
B
2
C
2
;
B
C
1
A
1
BC_1A_1
B
C
1
A
1
and
B
C
2
A
2
BC_2A_2
B
C
2
A
2
;
C
A
1
B
1
CA_1B_1
C
A
1
B
1
and
C
A
2
B
2
CA_2B_2
C
A
2
B
2
. Prove that
d
A
d_A
d
A
,
d
B
d_B
d
B
and
d
C
d_C
d
C
are concurrent if and only if \alpha a \plus{} \beta b \plus{} \gamma c \neq 0.
Find greatest possible length k of a circuit in such a graph
A graph has
n
n
n
vertices and \frac {1}{2}\left(n^2 \minus{} 3n \plus{} 4\right) edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length
k
k
k
of a circuit in such a graph.