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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1967 Vietnam National Olympiad
1967 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
1
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graph of | x^3 - x^2 - 2x | / 3 - | x + 1 |
Draw the graph of the function
y
=
∣
x
3
−
x
2
−
2
x
∣
3
−
∣
x
+
1
∣
y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |
y
=
3
∣
x
3
−
x
2
−
2
x
∣
−
∣
x
+
1∣
.
3
1
Hide problems
old Vietnamese rhombus
i)
A
B
C
D
ABCD
A
BC
D
is a rhombus. A tangent to the inscribed circle meets
A
B
,
D
A
,
B
C
,
C
D
AB, DA, BC, CD
A
B
,
D
A
,
BC
,
C
D
at
M
,
N
,
P
,
Q
M, N, P, Q
M
,
N
,
P
,
Q
respectively. Find a relationship between
B
M
BM
BM
and
D
N
DN
D
N
. ii)
A
B
C
D
ABCD
A
BC
D
is a rhombus and
P
P
P
a point inside. The circles through
P
P
P
with centers
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
meet the four sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon. iii) Rotate the figure about the line
A
C
AC
A
C
to form a solid. State a similar result.
2
1
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VietNam MO 1966-Pr.2
A river flows at speed u. A boat has speed v relative to the water. If its velocity is at an angle
α
\alpha
α
relative the direction of the river, what is its speed relative to the river bank? What
α
\alpha
α
minimises the time taken to cross the river?