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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1982 Vietnam National Olympiad
1982 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
Equality of areas for equilateral triangles on sides...
Let be given a triangle
A
B
C
ABC
A
BC
. Equilateral triangles
B
C
A
1
BCA_1
BC
A
1
and
B
C
A
2
BCA_2
BC
A
2
are drawn so that
A
A
A
and
A
1
A_1
A
1
are on one side of
B
C
BC
BC
, whereas
A
2
A_2
A
2
is on the other side. Points
B
1
,
B
2
,
C
1
,
C
2
B_1,B_2,C_1,C_2
B
1
,
B
2
,
C
1
,
C
2
are analogously defined. Prove that
S
A
B
C
+
S
A
1
B
1
C
1
=
S
A
2
B
2
C
2
.
S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.
S
A
BC
+
S
A
1
B
1
C
1
=
S
A
2
B
2
C
2
.
No line intersects four lines in a cube ABCDA'B'C'D'
Let
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
be a cube (where
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
are faces and
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA',BB',CC',DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
are edges). Consider the four lines
A
A
′
,
B
C
,
D
′
C
′
AA', BC, D'C'
A
A
′
,
BC
,
D
′
C
′
and the line joining the midpoints of
B
B
′
BB'
B
B
′
and
D
D
′
DD'
D
D
′
. Show that there is no line which cuts all the four lines.
2
2
Hide problems
Solve for x: x(x + 1)(x + 2)(x + 3) + 1 - m = 0.
For a given parameter
m
m
m
, solve the equation
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
+
1
−
m
=
0.
x(x + 1)(x + 2)(x + 3) + 1 - m = 0.
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
+
1
−
m
=
0.
Inequality in three variables and product.
Let
p
p
p
be a positive integer and
q
,
z
q, z
q
,
z
be real numbers with 0\le q\le 1 and q^{p+1}\le z\le 1. Prove that
∏
k
=
1
p
∣
z
−
q
k
z
+
q
k
∣
≤
∏
k
=
1
p
∣
1
−
q
k
1
+
q
k
∣
.
\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.
k
=
1
∏
p
z
+
q
k
z
−
q
k
≤
k
=
1
∏
p
1
+
q
k
1
−
q
k
.
1
2
Hide problems
Roots of polynomial are cos 72 degrees and cos 144 degrees.
Determine a quadric polynomial with intergral coefficients whose roots are
cos
7
2
∘
\cos 72^{\circ}
cos
7
2
∘
and
cos
14
4
∘
.
\cos 144^{\circ}.
cos
14
4
∘
.
number theory equation
Find all positive integers
x
,
y
,
z
x, y, z
x
,
y
,
z
such that
2
x
+
2
y
+
2
z
=
2336
2^x + 2^y + 2^z = 2336
2
x
+
2
y
+
2
z
=
2336
.