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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1989 Vietnam National Olympiad
1989 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
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A square of side length 2
A square
A
B
C
D
ABCD
A
BC
D
of side length
2
2
2
is given on a plane. The segment
A
B
AB
A
B
is moved continuously towards
C
D
CD
C
D
until
A
A
A
and
C
C
C
coincide with
C
C
C
and
D
D
D
, respectively. Let
S
S
S
be the area of the region formed by the segment
A
B
AB
A
B
while moving. Prove that
A
B
AB
A
B
can be moved in such a way that
S
<
5
π
6
S <\frac{5\pi}{6}
S
<
6
5
π
.
Parallelepiped and a line
Let be given a parallelepiped
A
B
C
D
.
A
′
B
′
C
′
D
′
ABCD.A'B'C'D'
A
BC
D
.
A
′
B
′
C
′
D
′
. Show that if a line
Δ
\Delta
Δ
intersects three of the lines
A
B
′
AB'
A
B
′
,
B
C
′
BC'
B
C
′
,
C
D
′
CD'
C
D
′
,
D
A
′
DA'
D
A
′
, then it intersects also the fourth line.
2
2
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Fibonacci sequence
The Fibonacci sequence is defined by F_1 \equal{} F_2 \equal{} 1 and F_{n\plus{}1} \equal{} F_n \plus{}F_{n\minus{}1} for
n
>
1
n > 1
n
>
1
. Let f(x) \equal{} 1985x^2 \plus{} 1956x \plus{} 1960. Prove that there exist infinitely many natural numbers
n
n
n
for which
f
(
F
n
)
f(F_n)
f
(
F
n
)
is divisible by
1989
1989
1989
. Does there exist
n
n
n
for which f(F_n) \plus{} 2 is divisible by
1989
1989
1989
?
Sequence of polynomials
The sequence of polynomials \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty} is defined inductively by P_0(x) \equal{} 0 and P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}. Prove that for any
x
∈
[
0
,
1
]
x \in [0, 1]
x
∈
[
0
,
1
]
and any natural number
n
n
n
it holds that 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}.
1
2
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Inequality
Let
n
n
n
and
N
N
N
be natural number. Prove that for any
α
\alpha
α
,
0
≤
α
≤
N
0\le\alpha\le N
0
≤
α
≤
N
, and any real
x
x
x
, it holds that { |\sum_ {k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}
Integer equation
Are there integers
x
x
x
,
y
y
y
, not both divisible by
5
5
5
, such that x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}?