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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1984 Vietnam National Olympiad
1984 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
Finding points and loci for a given square...
A square
A
B
C
D
ABCD
A
BC
D
of side length
2
a
2a
2
a
is given on a plane
Π
\Pi
Π
. Let
S
S
S
be a point on the ray
A
x
Ax
A
x
perpendicular to
Π
\Pi
Π
such that
A
S
=
2
a
.
AS = 2a.
A
S
=
2
a
.
(
a
)
(a)
(
a
)
Let
M
∈
B
C
M \in BC
M
∈
BC
and
N
∈
C
D
N \in CD
N
∈
C
D
be two variable points.
i
i
i
. Find the positions of
M
,
N
M,N
M
,
N
such that
B
M
+
D
N
≥
3
2
BM + DN \ge \frac{3}{2}
BM
+
D
N
≥
2
3
, planes
S
A
M
SAM
S
A
M
and
S
M
N
SMN
SMN
are perpendicular and
B
M
⋅
D
N
BM \cdot DN
BM
⋅
D
N
is minimum.
i
i
ii
ii
. Find
M
M
M
and
N
N
N
such that
∠
M
A
N
=
4
5
∘
\angle MAN = 45^{\circ}
∠
M
A
N
=
4
5
∘
and the volume of
S
A
M
N
SAMN
S
A
MN
attains an extremum value. Find these values.
(
b
)
(b)
(
b
)
Let
Q
Q
Q
be a point such that
∠
A
Q
B
=
∠
A
Q
D
=
9
0
∘
\angle AQB = \angle AQD = 90^{\circ}
∠
A
QB
=
∠
A
Q
D
=
9
0
∘
. The line
D
Q
DQ
D
Q
intersects the plane
π
\pi
π
through
A
B
AB
A
B
perpendicular to
Π
\Pi
Π
at
Q
′
Q'
Q
′
.
i
i
i
. Find the locus of
Q
′
Q'
Q
′
.
i
i
ii
ii
. Let
K
K
K
be the locus of points
Q
Q
Q
and let
C
Q
CQ
CQ
meet
K
K
K
again at
R
R
R
. Let
D
R
DR
D
R
meets
Π
\Pi
Π
at
R
′
R'
R
′
. Prove that
s
i
n
2
∠
Q
′
D
B
+
s
i
n
2
∠
R
′
D
B
sin^2 \angle Q'DB + sin^2 \angle R'DB
s
i
n
2
∠
Q
′
D
B
+
s
i
n
2
∠
R
′
D
B
is independent of
Q
Q
Q
.
Showing trigonometric relation and angle relation.
Consider a trihedral angle
S
x
y
z
Sxyz
S
x
yz
with \angle xSz = \alpha , \angle xSy = \beta and
∠
y
S
z
=
γ
\angle ySz =\gamma
∠
y
S
z
=
γ
. Let
A
,
B
,
C
A,B,C
A
,
B
,
C
denote the dihedral angles at edges
y
,
z
,
x
y, z, x
y
,
z
,
x
respectively.
(
a
)
(a)
(
a
)
Prove that
sin
α
sin
A
=
sin
β
sin
B
=
sin
γ
sin
C
\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}
s
i
n
A
s
i
n
α
=
s
i
n
B
s
i
n
β
=
s
i
n
C
s
i
n
γ
(
b
)
(b)
(
b
)
Show that
α
+
β
=
18
0
∘
\alpha + \beta = 180^{\circ}
α
+
β
=
18
0
∘
if and only if
∠
A
+
∠
B
=
18
0
∘
.
\angle A + \angle B = 180^{\circ}.
∠
A
+
∠
B
=
18
0
∘
.
(
c
)
(c)
(
c
)
Assume that \alpha=\beta =\gamma = 90^{\circ}. Let
O
∈
S
z
O \in Sz
O
∈
S
z
be a fixed point such that
S
O
=
a
SO = a
SO
=
a
and let
M
,
N
M,N
M
,
N
be variable points on
x
,
y
x, y
x
,
y
respectively. Prove that
∠
S
O
M
+
∠
S
O
N
+
∠
M
O
N
\angle SOM +\angle SON +\angle MON
∠
SOM
+
∠
SON
+
∠
MON
is constant and find the locus of the incenter of
O
S
M
N
OSMN
OSMN
.
2
2
Hide problems
A sequence satisfying linear recurrence and limit.
The sequence
(
u
n
)
(u_n)
(
u
n
)
is defined by
u
1
=
1
,
u
2
=
2
u_1 = 1, u_2 = 2
u
1
=
1
,
u
2
=
2
and
u
n
+
1
=
3
u
n
−
u
n
−
1
u_{n+1} = 3u_n - u_{n-1}
u
n
+
1
=
3
u
n
−
u
n
−
1
for
n
≥
2
n \ge 2
n
≥
2
. Set
v
n
=
∑
k
=
1
n
arccot
u
k
v_n =\sum_{k=1}^n \text{arccot }u_k
v
n
=
∑
k
=
1
n
arccot
u
k
. Compute
lim
n
→
∞
v
n
\lim_{n\to\infty} v_n
lim
n
→
∞
v
n
.
Polynomial satisfying xP(x - a) = (x - b)P(x) for given a,b.
Given two real numbers
a
,
b
a, b
a
,
b
with
a
≠
0
a \neq 0
a
=
0
, find all polynomials
P
(
x
)
P(x)
P
(
x
)
which satisfy
x
P
(
x
−
a
)
=
(
x
−
b
)
P
(
x
)
.
xP(x - a) = (x - b)P(x).
x
P
(
x
−
a
)
=
(
x
−
b
)
P
(
x
)
.
1
2
Hide problems
Find polynomial with irrational root and solve an equation.
(
a
)
(a)
(
a
)
Find a polynomial with integer coefficients of the smallest degree having
2
+
3
3
\sqrt{2} + \sqrt[3]{3}
2
+
3
3
as a root.
(
b
)
(b)
(
b
)
Solve 1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}.
Finding minimum value of expression and solving an equation.
(
a
)
(a)
(
a
)
Let
x
,
y
x, y
x
,
y
be integers, not both zero. Find the minimum possible value of
∣
5
x
2
+
11
x
y
−
5
y
2
∣
|5x^2 + 11xy - 5y^2|
∣5
x
2
+
11
x
y
−
5
y
2
∣
.
(
b
)
(b)
(
b
)
Find all positive real numbers
t
t
t
such that
9
t
10
=
[
t
]
t
−
[
t
]
\frac{9t}{10}=\frac{[t]}{t - [t]}
10
9
t
=
t
−
[
t
]
[
t
]
.