MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1998 Vietnam Team Selection Test
1998 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
sum of primes fractions > ln(ln(m))
Let
p
(
1
)
,
p
(
2
)
,
…
,
p
(
k
)
p(1), p(2), \ldots, p(k)
p
(
1
)
,
p
(
2
)
,
…
,
p
(
k
)
be all primes smaller than
m
m
m
, prove that
∑
i
=
1
k
1
p
(
i
)
+
1
p
(
i
)
2
>
l
n
(
l
n
(
m
)
)
.
\sum^{k}_{i=1} \frac{1}{p(i)} + \frac{1}{p(i)^2} > ln(ln(m)).
i
=
1
∑
k
p
(
i
)
1
+
p
(
i
)
2
1
>
l
n
(
l
n
(
m
))
.
There doesn't exist a Hamilton path
In a conference there are
n
≥
10
n \geq 10
n
≥
10
people. It is known that: I. Each person knows at least
[
n
+
2
3
]
\left[\frac{n+2}{3}\right]
[
3
n
+
2
]
other people. II. For each pair of person
A
A
A
and
B
B
B
who don't know each other, there exist some people
A
(
1
)
,
A
(
2
)
,
…
,
A
(
k
)
A(1), A(2), \ldots, A(k)
A
(
1
)
,
A
(
2
)
,
…
,
A
(
k
)
such that
A
A
A
knows
A
(
1
)
A(1)
A
(
1
)
,
A
(
i
)
A(i)
A
(
i
)
knows
A
(
i
+
1
)
A(i+1)
A
(
i
+
1
)
and
A
(
k
)
A(k)
A
(
k
)
knows
B
B
B
. III. There doesn't exist a Hamilton path. Prove that: We can divide those people into 2 groups:
A
A
A
group has a Hamilton cycle, and the other contains of people who don't know each other.
2
2
Hide problems
lots of tangent circles
In the plane we are given the circles
Γ
\Gamma
Γ
and
Δ
\Delta
Δ
tangent to each other and
Γ
\Gamma
Γ
contains
Δ
\Delta
Δ
. The radius of
Γ
\Gamma
Γ
is
R
R
R
and of
Δ
\Delta
Δ
is
R
2
\frac{R}{2}
2
R
. Prove that for each positive integer
n
≥
3
n \geq 3
n
≥
3
, the equation:
(
p
(
1
)
−
p
(
n
)
)
2
=
(
n
−
1
)
2
⋅
(
2
⋅
(
p
(
1
)
+
p
(
n
)
)
−
(
n
−
1
)
2
−
8
)
(p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8)
(
p
(
1
)
−
p
(
n
)
)
2
=
(
n
−
1
)
2
⋅
(
2
⋅
(
p
(
1
)
+
p
(
n
))
−
(
n
−
1
)
2
−
8
)
is the necessary and sufficient condition for
n
n
n
to exist
n
n
n
distinct circles
Υ
1
,
Υ
2
,
…
,
Υ
n
\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n
Υ
1
,
Υ
2
,
…
,
Υ
n
such that all these circles are tangent to
Γ
\Gamma
Γ
and
Δ
\Delta
Δ
and
Υ
i
\Upsilon_i
Υ
i
is tangent to
Υ
i
+
1
\Upsilon_{i+1}
Υ
i
+
1
, and
Υ
1
\Upsilon_1
Υ
1
has radius
R
p
(
1
)
\frac{R}{p(1)}
p
(
1
)
R
and
Υ
n
\Upsilon_n
Υ
n
has radius
R
p
(
n
)
\frac{R}{p(n)}
p
(
n
)
R
.
let d be a positive divisor of 5 + 1998^{1998}
Let
d
d
d
be a positive divisor of
5
+
199
8
1998
5 + 1998^{1998}
5
+
199
8
1998
. Prove that
d
=
2
⋅
x
2
+
2
⋅
x
⋅
y
+
3
⋅
y
2
d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2
d
=
2
⋅
x
2
+
2
⋅
x
⋅
y
+
3
⋅
y
2
, where
x
,
y
x, y
x
,
y
are integers if and only if
d
d
d
is congruent to 3 or 7
(
m
o
d
20
)
\pmod{20}
(
mod
20
)
.
1
2
Hide problems
| f(x) - P(x)| <= c * x^{1998}
Let
f
(
x
)
f(x)
f
(
x
)
be a real function such that for each positive real
c
c
c
there exist a polynomial
P
(
x
)
P(x)
P
(
x
)
(maybe dependent on
c
c
c
) such that
∣
f
(
x
)
−
P
(
x
)
∣
≤
c
⋅
x
1998
| f(x) - P(x)| \leq c \cdot x^{1998}
∣
f
(
x
)
−
P
(
x
)
∣
≤
c
⋅
x
1998
for all real
x
x
x
. Prove that
f
f
f
is a real polynomial.
p(a) is a positive integer
Find all integer polynomials
P
(
x
)
P(x)
P
(
x
)
, the highest coefficent is 1 such that: there exist infinitely irrational numbers
a
a
a
such that
p
(
a
)
p(a)
p
(
a
)
is a positive integer.