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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1993 Vietnam Team Selection Test
1993 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
maximal and the minimal values of x_1 - 2 * x_2 + x_3
Let's consider the real numbers
x
1
,
x
2
,
x
3
,
x
4
x_1, x_2, x_3, x_4
x
1
,
x
2
,
x
3
,
x
4
satisfying the condition
1
2
≤
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
≤
1
\dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1
2
1
≤
x
1
2
+
x
2
2
+
x
3
2
+
x
4
2
≤
1
Find the maximal and the minimal values of expression:
A
=
(
x
1
−
2
⋅
x
2
+
x
3
)
2
+
(
x
2
−
2
⋅
x
3
+
x
4
)
2
+
(
x
2
−
2
⋅
x
1
)
2
+
(
x
3
−
2
⋅
x
4
)
2
A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2
A
=
(
x
1
−
2
⋅
x
2
+
x
3
)
2
+
(
x
2
−
2
⋅
x
3
+
x
4
)
2
+
(
x
2
−
2
⋅
x
1
)
2
+
(
x
3
−
2
⋅
x
4
)
2
maximal value of n for which we can paint all edges
Let
n
n
n
points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
, (
n
>
2
n>2
n
>
2
), be considered in the space, where no four points are coplanar. Each pair of points
A
i
,
A
j
A_i, A_j
A
i
,
A
j
are connected by an edge. Find the maximal value of
n
n
n
for which we can paint all edges by two colors – blue and red such that the following three conditions hold: I. Each edge is painted by exactly one color. II. For
i
=
1
,
2
,
…
,
n
i = 1, 2, \ldots, n
i
=
1
,
2
,
…
,
n
, the number of blue edges with one end
A
i
A_i
A
i
does not exceed 4. III. For every red edge
A
i
A
j
A_iA_j
A
i
A
j
, we can find at least one point
A
k
A_k
A
k
(
k
≠
i
,
j
k \neq i, j
k
=
i
,
j
) such that the edges
A
i
A
k
A_iA_k
A
i
A
k
and
A
j
A
k
A_jA_k
A
j
A
k
are blue.
2
2
Hide problems
find real numbers s.t. sequence has finite limit
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined by:
a
1
=
1
,
a
n
+
1
=
a
n
+
1
a
n
a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}
a
1
=
1
,
a
n
+
1
=
a
n
+
a
n
1
for
n
=
1
,
2
,
3
,
…
n = 1, 2, 3, \ldots
n
=
1
,
2
,
3
,
…
. Find all real numbers
q
q
q
such that the sequence
{
u
n
}
\{u_n\}
{
u
n
}
defined by
u
n
=
a
n
q
u_n = a_n^q
u
n
=
a
n
q
,
n
=
1
,
2
,
3
,
…
n = 1, 2, 3, \ldots
n
=
1
,
2
,
3
,
…
has nonzero finite limit when
n
n
n
goes to infinity. THERE MIGHT BE A TYPO!
product with prime divisors of n
Let an integer
k
>
1
k > 1
k
>
1
be given. For each integer
n
>
1
n > 1
n
>
1
, we put
f
(
n
)
=
k
⋅
n
⋅
(
1
−
1
p
1
)
⋅
(
1
−
1
p
2
)
⋯
(
1
−
1
p
r
)
f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)
f
(
n
)
=
k
⋅
n
⋅
(
1
−
p
1
1
)
⋅
(
1
−
p
2
1
)
⋯
(
1
−
p
r
1
)
where
p
1
,
p
2
,
…
,
p
r
p_1, p_2, \ldots, p_r
p
1
,
p
2
,
…
,
p
r
are all distinct prime divisors of
n
n
n
. Find all values
k
k
k
for which the sequence
{
x
m
}
\{x_m\}
{
x
m
}
defined by
x
0
=
a
x_0 = a
x
0
=
a
and
x
m
+
1
=
f
(
x
m
)
,
m
=
0
,
1
,
2
,
3
,
…
x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots
x
m
+
1
=
f
(
x
m
)
,
m
=
0
,
1
,
2
,
3
,
…
is bounded for all integers
a
>
1
a > 1
a
>
1
.
1
2
Hide problems
orthocenter, incenter and circumcenter of triangle revisited
Let
H
H
H
,
I
I
I
,
O
O
O
be the orthocenter, incenter and circumcenter of a triangle. Show that
2
⋅
I
O
≥
I
H
2 \cdot IO \geq IH
2
⋅
I
O
≥
I
H
. When does the equality hold ?
2 x 3 rectangle tilings
We call a rectangle of size
2
×
3
2 \times 3
2
×
3
(or
3
×
2
3 \times 2
3
×
2
) without one cell in corner a
P
P
P
-rectangle. We call a rectangle of size
2
×
3
2 \times 3
2
×
3
(or
3
×
2
3 \times 2
3
×
2
) without two cells in opposite (under center of rectangle) corners a
S
S
S
-rectangle. Using some squares of size
2
×
2
2 \times 2
2
×
2
, some
P
P
P
-rectangles and some
S
S
S
-rectangles, one form one rectangle of size
1993
×
2000
1993 \times 2000
1993
×
2000
(figures don’t overlap each other). Let
s
s
s
denote the sum of numbers of squares and
S
S
S
-rectangles used in such tiling. Find the maximal value of
s
s
s
.