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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1981 Vietnam National Olympiad
1981 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
2
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Solving a system of four equations in four variables.
Solve the system of equations
x
2
+
y
2
+
z
2
+
t
2
=
50
;
x^2 + y^2 + z^2 + t^2 = 50;
x
2
+
y
2
+
z
2
+
t
2
=
50
;
x
2
−
y
2
+
z
2
−
t
2
=
−
24
;
x^2 - y^2 + z^2 - t^2 = -24;
x
2
−
y
2
+
z
2
−
t
2
=
−
24
;
x
y
=
z
t
;
xy = zt;
x
y
=
z
t
;
x
−
y
+
z
−
t
=
0.
x - y + z - t = 0.
x
−
y
+
z
−
t
=
0.
ABC is right triangled iff cyc sum sin A=1+cyc sum cos A
Prove that a triangle
A
B
C
ABC
A
BC
is right-angled if and only if
sin
A
+
sin
B
+
sin
C
=
cos
A
+
cos
B
+
cos
C
+
1
\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1
sin
A
+
sin
B
+
sin
C
=
cos
A
+
cos
B
+
cos
C
+
1
2
2
Hide problems
find all m
Consider the polynomials
f
(
p
)
=
p
12
−
p
11
+
3
p
10
+
11
p
3
−
p
2
+
23
p
+
30
;
f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;
f
(
p
)
=
p
12
−
p
11
+
3
p
10
+
11
p
3
−
p
2
+
23
p
+
30
;
g
(
p
)
=
p
3
+
2
p
+
m
.
g(p) = p^3 + 2p + m.
g
(
p
)
=
p
3
+
2
p
+
m
.
Find all integral values of
m
m
m
for which
f
f
f
is divisible by
g
g
g
.
A^2/B >= 4pq/(p+q)^2.
Let
p
,
q
p, q
p
,
q
be real numbers with
0
<
p
<
q
0 < p < q
0
<
p
<
q
and let
t
1
,
t
2
,
⋯
,
t
n
t_1, t_2, \cdots, t_n
t
1
,
t
2
,
⋯
,
t
n
be real numbers in the interval
[
p
,
q
]
[p, q]
[
p
,
q
]
. Denote by
A
A
A
and
B
B
B
the arithmetic means of
t
1
,
t
2
,
⋯
,
t
n
t_1, t_2, \cdots, t_n
t
1
,
t
2
,
⋯
,
t
n
and of
t
1
2
,
t
2
2
,
⋯
,
t
n
2
t_1^2, t_2^2,\cdots , t_n^2
t
1
2
,
t
2
2
,
⋯
,
t
n
2
, respectively. Prove that
A
2
B
≥
4
p
q
(
p
+
q
)
2
.
\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.
B
A
2
≥
(
p
+
q
)
2
4
pq
.
3
2
Hide problems
Determining a point on plane with ratio of distances minimal
A plane
ρ
\rho
ρ
and two points
M
,
N
M, N
M
,
N
outside it are given. Determine the point
A
A
A
on
ρ
\rho
ρ
for which
A
M
A
N
\frac{AM}{AN}
A
N
A
M
is minimal.
older geometry
Two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively touch externally at
A
A
A
. Let
M
M
M
be a point inside
k
2
k_2
k
2
and outside the line
O
1
O
2
O_1O_2
O
1
O
2
. Find a line
d
d
d
through
M
M
M
which intersects
k
1
k_1
k
1
and
k
2
k_2
k
2
again at
B
B
B
and
C
C
C
respectively so that the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
is tangent to
O
1
O
2
O_1O_2
O
1
O
2
.