MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1987 Vietnam National Olympiad
1987 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
2
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arithmetic progression
Let
u
1
u_1
u
1
,
u
2
u_2
u
2
,
…
\ldots
…
,
u
1987
u_{1987}
u
1987
be an arithmetic progression with u_1 \equal{} \frac {\pi}{1987} and the common difference
π
3974
\frac {\pi}{3974}
3974
π
. Evaluate S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right)
Inequality
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
n
a_n
a
n
be positive real numbers (
n
≥
2
n \ge 2
n
≥
2
) whose sum is
S
S
S
. Show that \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}} for any nonnegative integers
k
k
k
,
t
t
t
with
k
≥
t
k \ge t
k
≥
t
. When does equality occur?
2
2
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two sequences
Sequences
(
x
n
)
(x_n)
(
x
n
)
and
(
y
n
)
(y_n)
(
y
n
)
are constructed as follows: x_0 \equal{} 365, x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622, and y_0 \equal{} 16, y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952, for all
n
≥
0
n \ge 0
n
≥
0
. Prove that \left|x_n\minus{} y_k\right|\neq 0 for any positive integers
n
n
n
,
k
k
k
.
Differentiable function
Let f : [0, \plus{}\infty) \to \mathbb R be a differentiable function. Suppose that
∣
f
(
x
)
∣
≤
5
\left|f(x)\right| \le 5
∣
f
(
x
)
∣
≤
5
and
f
(
x
)
f
′
(
x
)
≥
sin
x
f(x)f'(x) \ge \sin x
f
(
x
)
f
′
(
x
)
≥
sin
x
for all
x
≥
0
x \ge 0
x
≥
0
. Prove that there exists \lim_{x\to\plus{}\infty}f(x).
3
2
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n points on a plane
Let be given
n
≥
2
n \ge 2
n
≥
2
lines on a plane, not all concurrent and no two parallel. Prove that there is a point which belongs to exactly two of the given lines.
Five rays in space
Prove that among any five distinct rays
O
x
Ox
O
x
,
O
y
Oy
O
y
,
O
z
Oz
O
z
,
O
t
Ot
Ot
,
O
r
Or
O
r
in space there exist two which form an angle less than or equal to
9
0
∘
90^{\circ}
9
0
∘
.