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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1999 Vietnam National Olympiad
1999 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
1
2
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System of equations
Solve the system of equations: (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1} y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0
Unusal inequality
Given are three positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying abc \plus{} a \plus{} c \equal{} b. Find the max value of the expression: P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.
3
1
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Count all the functions
Let S \equal{} \{0,1,2,\ldots,1999\} and T \equal{} \{0,1,2,\ldots \}. Find all functions
f
:
T
↦
S
f: T \mapsto S
f
:
T
↦
S
such that (i) f(s) \equal{} s \forall s \in S. (ii) f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \forall m,n \in T.
2
2
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equal segments and equilateral triangle
let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at
A
1
,
A
2
A_1,A_2
A
1
,
A
2
respectively. Pairs of point
(
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 similarly defined. Prove that A_1A_2 \equal{} B_1B_2 \equal{} C_1C_2 if and only if triangle ABC is equilateral.
vietnam 1999
O
A
,
O
B
,
O
C
,
O
D
OA, OB, OC, OD
O
A
,
OB
,
OC
,
O
D
are 4 rays in space such that the angle between any two is the same. Show that for a variable ray
O
X
,
OX,
OX
,
the sum of the cosines of the angles
X
O
A
,
X
O
B
,
X
O
C
,
X
O
D
XOA, XOB, XOC, XOD
XO
A
,
XOB
,
XOC
,
XO
D
is constant and the sum of the squares of the cosines is also constant.