MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1985 Vietnam National Olympiad
1985 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
2
Hide problems
Find all pairs of integers
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers such that x^3 \minus{} y^3 \equal{} 2xy \plus{} 8.
Divisible problem
Let
a
a
a
,
b
b
b
and
m
m
m
be positive integers. Prove that there exists a positive integer
n
n
n
such that (a^n \minus{} 1)b is divisible by
m
m
m
if and only if \gcd (ab, m) \equal{} \gcd (b, m).
3
2
Hide problems
Calculate the area
A parallelepiped with the side lengths
a
a
a
,
b
b
b
,
c
c
c
is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.
calculate the volume of the pyramid
A triangular pyramid
O
.
A
B
C
O.ABC
O
.
A
BC
with base
A
B
C
ABC
A
BC
has the property that the lengths of the altitudes from
A
A
A
,
B
B
B
and
C
C
C
are not less than \frac{OB \plus{}OC}{2}, \frac{OC \plus{} OA}{2} and \frac{OA \plus{} OB}{2}, respectively. Given that the area of
A
B
C
ABC
A
BC
is
S
S
S
, calculate the volume of the pyramid.
2
2
Hide problems
Find all functions
Find all functions
f
:
Z
↦
R
f \colon \mathbb{Z} \mapsto \mathbb{R}
f
:
Z
↦
R
which satisfy: i) f(x)f(y) \equal{} f(x \plus{} y) \plus{} f(x \minus{} y) for all integers
x
x
x
,
y
y
y
ii)
f
(
0
)
≠
0
f(0) \neq 0
f
(
0
)
=
0
iii) f(1) \equal{} \frac {5}{2}
Four solutions form an arithmetic progression
Find all real values of parameter
a
a
a
for which the equation in
x
x
x
16x^4 \minus{} ax^3 \plus{} (2a \plus{} 17)x^2 \minus{} ax \plus{} 16 \equal{} 0 has four solutions which form an arithmetic progression.