MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1990 Vietnam Team Selection Test
1990 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
f(f(x)) = x^2 - 2
Prove that there is no real function
f
(
x
)
f(x)
f
(
x
)
satisfying f\left(f(x)\right) \equal{} x^2 \minus{} 2 for all real number
x
x
x
.
Always face the teacher that stand at the circle centre
There are
n
≥
3
n\geq 3
n
≥
3
pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number
M
M
M
such that after
M
M
M
times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils
A
A
A
and
B
B
B
, and if
A
A
A
initially stands on the left side of
B
B
B
then
A
A
A
will finally stand on the right side of
B
B
B
.
2
2
Hide problems
Inequality in a tetrahedron
Given a tetrahedron such that product of the opposite edges is
1
1
1
. Let the angle between the opposite edges be
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
, and circumradii of four faces be
R
1
R_1
R
1
,
R
2
R_2
R
2
,
R
3
R_3
R
3
,
R
4
R_4
R
4
. Prove that \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}}
Find limit
Let be given four positive real numbers
a
a
a
,
b
b
b
,
A
A
A
,
B
B
B
. Consider a sequence of real numbers
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
,
…
\ldots
…
is given by x_1 \equal{} a, x_2 \equal{} b and x_{n \plus{} 1} \equal{} A\sqrt [3]{x_n^2} \plus{} B\sqrt [3]{x_{n \minus{} 1}^2} ( n \equal{} 2, 3, 4, \ldots). Prove that there exist limit \lim_{n\to \plus{} \propto}x_n and find this limit.
1
2
Hide problems
Inequality in a cyclic convex polygon
Let be given a convex polygon
M
0
M
1
…
M
2
n
M_0M_1\ldots M_{2n}
M
0
M
1
…
M
2
n
(
n
≥
1
n\ge 1
n
≥
1
), where 2n \plus{} 1 points
M
0
M_0
M
0
,
M
1
M_1
M
1
,
…
\ldots
…
,
M
2
n
M_{2n}
M
2
n
lie on a circle
(
C
)
(C)
(
C
)
with diameter
R
R
R
in an anticlockwise direction. Suppose that there is a point
A
A
A
inside this convex polygon such that
∠
M
0
A
M
1
\angle M_0AM_1
∠
M
0
A
M
1
,
∠
M
1
A
M
2
\angle M_1AM_2
∠
M
1
A
M
2
,
…
\ldots
…
, \angle M_{2n \minus{} 1}AM_{2n},
∠
M
2
n
A
M
0
\angle M_{2n}AM_0
∠
M
2
n
A
M
0
are equal. Assume that
A
A
A
is not coincide with the center of the circle
(
C
)
(C)
(
C
)
and
B
B
B
be a point lies on
(
C
)
(C)
(
C
)
such that
A
B
AB
A
B
is perpendicular to the diameter of
(
C
)
(C)
(
C
)
passes through
A
A
A
. Prove that \frac {2n \plus{} 1}{\frac {1}{AM_0} \plus{} \frac {1}{AM_1} \plus{} \cdots \plus{} \frac {1}{AM_{2n}}} < AB < \frac {AM_0 \plus{} AM_1 \plus{} \cdots \plus{} AM_{2n}}{2n \plus{} 1} < R
Set of positive integers
Let
T
T
T
be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of
T
T
T
, their greatest common divisor and their least common multiple are also elements of
T
T
T
. 2. For any element
x
x
x
of
T
T
T
, there exists an element
x
′
x'
x
′
of
T
T
T
such that
x
x
x
and
x
′
x'
x
′
are relatively prime, and their least common multiple is the largest number in
T
T
T
. For each such set
T
T
T
, denote by
s
(
T
)
s(T)
s
(
T
)
its number of elements. It is known that
s
(
T
)
<
1990
s(T) < 1990
s
(
T
)
<
1990
; find the largest value
s
(
T
)
s(T)
s
(
T
)
may take.