MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1996 Vietnam Team Selection Test
1996 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
(a+b)^4+(b+c)^4+(c+a)^4 - product
Find the minimum value of the expression:
f
(
a
,
b
,
c
)
=
(
a
+
b
)
4
+
(
b
+
c
)
4
+
(
c
+
a
)
4
−
4
7
⋅
(
a
4
+
b
4
+
c
4
)
.
f(a,b,c)= (a+b)^4+(b+c)^4+(c+a)^4 - \frac{4}{7} \cdot (a^4+b^4+c^4).
f
(
a
,
b
,
c
)
=
(
a
+
b
)
4
+
(
b
+
c
)
4
+
(
c
+
a
)
4
−
7
4
⋅
(
a
4
+
b
4
+
c
4
)
.
x_{n+1} = a / ( 1+x(n)^2)
Find all reals
a
a
a
such that the sequence
{
x
(
n
)
}
\{x(n)\}
{
x
(
n
)}
,
n
=
0
,
1
,
2
,
…
n=0,1,2, \ldots
n
=
0
,
1
,
2
,
…
that satisfy:
x
(
0
)
=
1996
x(0)=1996
x
(
0
)
=
1996
and
x
n
+
1
=
a
1
+
x
(
n
)
2
x_{n+1} = \frac{a}{1+x(n)^2}
x
n
+
1
=
1
+
x
(
n
)
2
a
for any natural number
n
n
n
has a limit as n goes to infinity.
2
2
Hide problems
2^f(n) divides sum of binomials
For each positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the maximal natural number such that:
2
f
(
n
)
2^{f(n)}
2
f
(
n
)
divides
∑
i
=
0
⌊
n
−
1
2
⌋
(
n
2
⋅
i
+
1
)
3
i
\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i
∑
i
=
0
⌊
2
n
−
1
⌋
(
2
⋅
i
+
1
n
)
3
i
. Find all
n
n
n
such that
f
(
n
)
=
1996.
f(n) = 1996.
f
(
n
)
=
1996.
[hide="old version"]For each positive integer
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the maximal natural number such that:
2
f
(
n
)
2^{f(n)}
2
f
(
n
)
divides
∑
i
=
1
n
+
1
/
2
(
2
⋅
i
+
1
n
)
\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}
∑
i
=
1
n
+
1/2
(
n
2
⋅
i
+
1
)
. Find all
n
n
n
such that
f
(
n
)
=
1996.
f(n) = 1996.
f
(
n
)
=
1996.
Can the number of people be 65?
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?
1
2
Hide problems
n pairwise disjoint triangles
In the plane we are given
3
⋅
n
3 \cdot n
3
⋅
n
points (
n
>
n>
n
>
1) no three collinear, and the distance between any two of them is
≤
1
\leq 1
≤
1
. Prove that we can construct
n
n
n
pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles
<
1
/
2
< 1/2
<
1/2
.
d does not depend on the order of the axes of symmetry
Given 3 non-collinear points
A
,
B
,
C
A,B,C
A
,
B
,
C
. For each point
M
M
M
in the plane (
A
B
C
ABC
A
BC
) let
M
1
M_1
M
1
be the point symmetric to
M
M
M
with respect to
A
B
AB
A
B
,
M
2
M_2
M
2
be the point symmetric to
M
1
M_1
M
1
with respect to
B
C
BC
BC
and
M
′
M'
M
′
be the point symmetric to
M
2
M_2
M
2
with respect to
A
C
AC
A
C
. Find all points
M
M
M
such that
M
M
′
MM'
M
M
′
obtains its minimum. Let this minimum value be
d
d
d
. Prove that
d
d
d
does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
. In the first part the order of axes we chose
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
, and the second part of the problem states that the value
d
d
d
doesn't depend on this order).