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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1999 Vietnam Team Selection Test
1999 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
clockwise sides of convex polygon
Let a convex polygon
H
H
H
be given. Show that for every real number
a
∈
(
0
,
1
)
a \in (0, 1)
a
∈
(
0
,
1
)
there exist 6 distinct points on the sides of
H
H
H
, denoted by
A
1
,
A
2
,
…
,
A
6
A_1, A_2, \ldots, A_6
A
1
,
A
2
,
…
,
A
6
clockwise, satisfying the conditions: I.
(
A
1
A
2
)
=
(
A
5
A
4
)
=
a
⋅
(
A
6
A
3
)
(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)
(
A
1
A
2
)
=
(
A
5
A
4
)
=
a
⋅
(
A
6
A
3
)
. II. Lines
A
1
A
2
,
A
5
A
4
A_1A_2, A_5A_4
A
1
A
2
,
A
5
A
4
are equidistant from
A
6
A
3
A_6A_3
A
6
A
3
. (By
(
A
B
)
(AB)
(
A
B
)
we denote vector
A
B
AB
A
B
)
Monkeys are going to pick up peanuts
Let a regular polygon with
p
p
p
vertices be given, where
p
p
p
is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes
p
p
p
peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the
k
k
k
-th peanut, he skips the
2
⋅
k
2 \cdot k
2
⋅
k
next vertices and gives
k
+
1
k+1
k
+
1
-th for the monkey at the next vertex. He does so until all
p
p
p
peanuts are delivered. I. How many monkeys are there which does not receive peanuts? II. How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?
2
2
Hide problems
real coefficients are called similar
Two polynomials
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
with real coefficients are called similar if there exist nonzero real number a such that
f
(
x
)
=
q
⋅
g
(
x
)
f(x) = q \cdot g(x)
f
(
x
)
=
q
⋅
g
(
x
)
for all
x
∈
R
x \in R
x
∈
R
. I. Show that there exists a polynomial
P
(
x
)
P(x)
P
(
x
)
of degree 1999 with real coefficients which satisfies the condition:
(
P
(
x
)
)
2
−
4
(P(x))^2 - 4
(
P
(
x
)
)
2
−
4
and
(
P
′
(
x
)
)
2
⋅
(
x
2
−
4
)
(P'(x))^2 \cdot (x^2-4)
(
P
′
(
x
)
)
2
⋅
(
x
2
−
4
)
are similar. II. How many polynomials of degree 1999 are there which have above mentioned property.
Circles touches side of triangle
Let a triangle
A
B
C
ABC
A
BC
inscribed in circle
Γ
\Gamma
Γ
be given. Circle
Θ
\Theta
Θ
lies in angle
A
^
Â
A
^
of triangle and touches sides
A
B
,
A
C
AB, AC
A
B
,
A
C
at
M
1
,
N
1
M_1, N_1
M
1
,
N
1
and touches internally
Γ
\Gamma
Γ
at
P
1
P_1
P
1
. The points
M
2
,
N
2
,
P
2
M_2, N_2, P_2
M
2
,
N
2
,
P
2
and
M
3
,
N
3
,
P
3
M_3, N_3, P_3
M
3
,
N
3
,
P
3
are defined similarly to angles
B
B
B
and
C
C
C
respectively. Show that
M
1
N
1
,
M
2
N
2
M_1N_1, M_2N_2
M
1
N
1
,
M
2
N
2
and
M
3
N
3
M_3N_3
M
3
N
3
intersect each other at their midpoints.
1
2
Hide problems
Vietnam TST 1999 sum inequality
Let a sequence of positive reals
{
u
n
}
n
=
1
∞
\{u_n\}^{\infty}_{n=1}
{
u
n
}
n
=
1
∞
be given. For every positive integer
n
n
n
, let
k
n
k_n
k
n
be the least positive integer satisfying:
∑
i
=
1
k
n
1
i
≥
∑
i
=
1
n
u
i
.
\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.
i
=
1
∑
k
n
i
1
≥
i
=
1
∑
n
u
i
.
Show that the sequence
{
k
n
+
1
k
n
}
\left\{\frac{k_{n+1}}{k_n}\right\}
{
k
n
k
n
+
1
}
has finite limit if and only if
{
u
n
}
\{u_n\}
{
u
n
}
does.
2^h <> 1 (mod p)
Let an odd prime
p
p
p
be a given number satisfying
2
h
≠
1
(
m
o
d
p
)
2^h \neq 1 \pmod{p}
2
h
=
1
(
mod
p
)
for all
h
<
p
−
1
,
h
∈
N
∗
,
h < p-1, h \in \mathbb{N}^{*},
h
<
p
−
1
,
h
∈
N
∗
,
and an even integer
a
∈
(
p
2
,
p
)
.
a \in \left(\frac{p}{2},p \right).
a
∈
(
2
p
,
p
)
.
Let us consider the sequence
{
a
n
}
n
=
0
∞
\{a_n\}^{\infty}_{n=0}
{
a
n
}
n
=
0
∞
defined by
a
0
=
a
a_0 = a
a
0
=
a
and
a
n
+
1
=
p
−
b
n
a_{n+1} = p - b_n
a
n
+
1
=
p
−
b
n
for
n
=
0
,
1
,
2
,
…
n = 0, 1, 2, \ldots
n
=
0
,
1
,
2
,
…
, where
b
n
b_n
b
n
is the greatest odd divisor of
a
n
.
a_n.
a
n
.
Show that
{
a
n
}
\{a_n\}
{
a
n
}
is periodical and find its least positive period.