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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1976 Vietnam National Olympiad
1976 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(6)
2
1
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(acosA + bcosB + ccosC)/(asinA + bsinB + csinC) = (a + b + c)/9R
Find all triangles
A
B
C
ABC
A
BC
such that
a
c
o
s
A
+
b
c
o
s
B
+
c
c
o
s
C
a
s
i
n
A
+
b
s
i
n
B
+
c
s
i
n
C
=
a
+
b
+
c
9
R
\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}
a
s
in
A
+
b
s
in
B
+
cs
in
C
a
cos
A
+
b
cos
B
+
ccos
C
=
9
R
a
+
b
+
c
, where, as usual,
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
and
R
R
R
is the circumradius.
5
1
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4 lines in spaces, find position of a line that minimises a distance of 2 lines
L
,
L
′
L, L'
L
,
L
′
are two skew lines in space and
p
p
p
is a plane not containing either line.
M
M
M
is a variable line parallel to
p
p
p
which meets
L
L
L
at
X
X
X
and
L
′
L'
L
′
at
Y
Y
Y
. Find the position of
M
M
M
which minimises the distance
X
Y
XY
X
Y
.
L
′
′
L''
L
′′
is another fixed line. Find the line
M
M
M
which is also perpendicular to
L
′
′
L''
L
′′
.
6
1
Hide problems
Σ1/x_i^n from i=1 to i=k \ge k^{n+1} when Σx_i=1
Show that
1
x
1
n
+
1
x
2
n
+
.
.
.
+
1
x
k
n
≥
k
n
+
1
\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}
x
1
n
1
+
x
2
n
1
+
...
+
x
k
n
1
≥
k
n
+
1
for positive real numbers
x
i
x_i
x
i
with sum
1
1
1
.
4
1
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find all 3 digit integers \overline{abc} = n so that 2n/3= a! b! c!
Find all three digit integers
a
b
c
‾
=
n
\overline{abc} = n
ab
c
=
n
, such that
2
n
3
=
a
!
b
!
c
!
\frac{2n}{3} = a! b! c!
3
2
n
=
a
!
b
!
c
!
3
1
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p<= 8S^3 / 27abc, where p is product of distances of P from sides of ABC, S=area
P
P
P
is a point inside the triangle
A
B
C
ABC
A
BC
. The perpendicular distances from
P
P
P
to the three sides have product
p
p
p
. Show that
p
≤
8
S
3
27
a
b
c
p \le \frac{ 8 S^3}{27abc}
p
≤
27
ab
c
8
S
3
, where
S
=
S =
S
=
area
A
B
C
ABC
A
BC
and
a
,
b
,
c
a, b, c
a
,
b
,
c
are the sides. Prove a similar result for a tetrahedron.
1
1
Hide problems
all integer solutions to m^{m+n} = n^{12} and n^{m+n} = m^3
Find all integer solutions to
m
m
+
n
=
n
12
,
n
m
+
n
=
m
3
m^{m+n} = n^{12}, n^{m+n} = m^3
m
m
+
n
=
n
12
,
n
m
+
n
=
m
3
.