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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1997 Vietnam National Olympiad
1997 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
2
2
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sequences
Let n be an integer which is greater than 1, not divisible by 1997. Let a_m\equal{}m\plus{}\frac{mn}{1997} for all m=1,2,..,1996 b_m\equal{}m\plus{}\frac{1997m}{n} for all m=1,2,..,n-1 We arrange the terms of two sequence
(
a
i
)
,
(
b
j
)
(a_i), (b_j)
(
a
i
)
,
(
b
j
)
in the ascending order to form a new sequence c_1\le c_2\le ...\le c_{1995\plus{}n} Prove that c_{k\plus{}1}\minus{}c_k<2 for all k=1,2,...,1994+n
2^n | 19^k-97
Prove that for evey positive integer n, there exits a positive integer k such that 2^n | 19^k \minus{} 97
3
2
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numbers of functions
Find the number of functions
f
:
N
→
N
f: \mathbb N\rightarrow\mathbb N
f
:
N
→
N
which satisfying: (i) f(1) \equal{} 1 (ii) f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997 for every natural numbers n.
one's area is not greater than 7/72
In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.
1
2
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Evaluate the maximum and minimum values
Given a circle (O,R). A point P lies inside the circle, OP=d, d
Polynomial function
Let k \equal{} \sqrt[3]{3}. a, Find all polynomials
p
(
x
)
p(x)
p
(
x
)
with rationl coefficients whose degree are as least as possible such that p(k \plus{} k^2) \equal{} 3 \plus{} k. b, Does there exist a polynomial
p
(
x
)
p(x)
p
(
x
)
with integer coefficients satisfying p(k \plus{} k^2) \equal{} 3 \plus{} k