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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1991 Vietnam National Olympiad
1991 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
Vietnam
Three mutually perpendicular rays
O
x
,
O
y
,
O
z
O_x,O_y,O_z
O
x
,
O
y
,
O
z
and three points
A
,
B
,
C
A,B,C
A
,
B
,
C
on
O
x
,
O
y
,
O
z
O_x,O_y,O_z
O
x
,
O
y
,
O
z
, respectively. A variable sphere є through
A
,
B
,
C
A, B,C
A
,
B
,
C
meets
O
x
,
O
y
,
O
z
O_x,O_y,O_z
O
x
,
O
y
,
O
z
again at
A
′
,
B
′
,
C
′
A', B',C'
A
′
,
B
′
,
C
′
, respectively. Let
M
M
M
and
M
′
M'
M
′
be the centroids of triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
. Find the locus of the midpoint of
M
M
′
MM'
M
M
′
.
VietNam MO 1991
Prove that: \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2} where
x
;
y
;
z
x;y;z
x
;
y
;
z
are real numbers saisfying
x
≥
y
≥
z
≥
0
x \geq y \geq z \geq 0
x
≥
y
≥
z
≥
0
2
2
Hide problems
2^f(n) divides k^n-1
Let
k
>
1
k>1
k
>
1
be an odd integer. For every positive integer n, let
f
(
n
)
f(n)
f
(
n
)
be the greatest positive integer for which
2
f
(
n
)
2^{f(n)}
2
f
(
n
)
divides
k
n
−
1
k^n-1
k
n
−
1
. Find
f
(
n
)
f(n)
f
(
n
)
in terms of
k
k
k
and
n
n
n
.
Vietnam
Let
G
G
G
be centroid and
R
R
R
the circunradius of a triangle
A
B
C
ABC
A
BC
. The extensions of
G
A
,
G
B
,
G
C
GA,GB,GC
G
A
,
GB
,
GC
meet the circuncircle again at
D
,
E
,
F
D,E,F
D
,
E
,
F
. Prove that:
3
R
≤
1
G
D
+
1
G
E
+
1
G
F
≤
3
≤
1
A
B
+
1
B
C
+
1
C
A
\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}
R
3
≤
G
D
1
+
GE
1
+
GF
1
≤
3
≤
A
B
1
+
BC
1
+
C
A
1
1
2
Hide problems
Vietnam
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
satisfying:
f
(
x
y
)
+
f
(
x
z
)
2
−
f
(
x
)
f
(
y
z
)
≥
1
4
\frac{f(xy)+f(xz)}{2} - f(x)f(yz) \geq \frac{1}{4}
2
f
(
x
y
)
+
f
(
x
z
)
−
f
(
x
)
f
(
yz
)
≥
4
1
for all
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
Vietnam
1991
1991
1991
students sit around a circle and play the following game. Starting from some student
A
A
A
and counting clockwise, each student on turn says a number. The numbers are
1
,
2
,
3
,
1
,
2
,
3
,
.
.
.
1,2,3,1,2,3,...
1
,
2
,
3
,
1
,
2
,
3
,
...
A student who says
2
2
2
or
3
3
3
must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?