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Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
1997 Vietnam Team Selection Test
1997 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
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Find the greatest real number...
Find the greatest real number
α
\alpha
α
for which there exists a sequence of infinitive integers
(
a
n
)
(a_n)
(
a
n
)
, ( n \equal{} 1, 2, 3, \ldots) satisfying the following conditions: 1)
a
n
>
1997
n
a_n > 1997n
a
n
>
1997
n
for every
n
∈
N
∗
n \in\mathbb{N}^{*}
n
∈
N
∗
; 2) For every
n
≥
2
n\ge 2
n
≥
2
,
U
n
≥
a
n
α
U_n\ge a^{\alpha}_n
U
n
≥
a
n
α
, where U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}.
Colors of points on a circle
Let
n
n
n
,
k
k
k
,
p
p
p
be positive integers with 2 \le k \le \frac {n}{p \plus{} 1}. Let
n
n
n
distinct points on a circle be given. These points are colored blue and red so that exactly
k
k
k
points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least
p
p
p
red points. How many such colorings are there?
2
2
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Flight routes in a country
There are
25
25
25
towns in a country. Find the smallest
k
k
k
for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied: 1) from each town there are exactly
k
k
k
direct routes to
k
k
k
other towns; 2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
Find all pairs of positive real numbers
Find all pairs of positive real numbers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that for every
n
∈
N
∗
n \in\mathbb{N}^*
n
∈
N
∗
and every real root
x
n
x_n
x
n
of the equation 4n^2x \equal{} \log_2(2n^2x \plus{} 1) we always have a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n.
1
2
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Arbitrary integers not smaller than k
The function
f
:
N
→
Z
f : \mathbb{N} \to \mathbb{Z}
f
:
N
→
Z
is defined by f(0) \equal{} 2, f(1) \equal{} 503 and f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n) for all
n
∈
N
n \in\mathbb{N}
n
∈
N
. Let
s
1
s_1
s
1
,
s
2
s_2
s
2
,
…
\ldots
…
,
s
k
s_k
s
k
be arbitrary integers not smaller than
k
k
k
, and let
p
(
s
i
)
p(s_i)
p
(
s
i
)
be an arbitrary prime divisor of
f
(
2
s
i
)
f\left(2^{s_i}\right)
f
(
2
s
i
)
, ( i \equal{} 1, 2, \ldots, k). Prove that, for any positive integer
t
t
t
(
t
≤
k
t\le k
t
≤
k
), we have 2^t \Big | \sum_{i \equal{} 1}^kp(s_i) if and only if
2
t
∣
k
2^t | k
2
t
∣
k
.
Prove that there is a unique point satisfying conditions
Let
A
B
C
D
ABCD
A
BC
D
be a given tetrahedron, with BC \equal{} a, CA \equal{} b, AB \equal{} c, DA \equal{} a_1, DB \equal{} b_1, DC \equal{} c_1. Prove that there is a unique point
P
P
P
satisfying PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2 and for this point
P
P
P
we have PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2, where
R
R
R
is the circumradius of the tetrahedron
A
B
C
D
ABCD
A
BC
D
. Find the necessary and sufficient condition so that this inequality is an equality.