MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2012 Hanoi Open Mathematics Competitions
8
8
Part of
2012 Hanoi Open Mathematics Competitions
Problems
(2)
Q8 - Hanoi Open Mathematical Olympiad 2012 - Junior Section
Source:
6/10/2012
Q8. Given a triangle
A
B
C
ABC
A
BC
and
2
2
2
point
K
∈
A
B
,
N
∈
B
C
K \in AB, \; N \in BC
K
∈
A
B
,
N
∈
BC
such that
B
K
=
2
A
K
,
C
N
=
2
B
N
BK=2AK, \; CN=2BN
B
K
=
2
A
K
,
CN
=
2
BN
and
Q
Q
Q
is the common point of
A
N
AN
A
N
and
C
K
CK
C
K
. Compute
S
△
A
B
C
S
△
B
C
Q
.
\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.
S
△
BCQ
S
△
A
BC
.
Determine the greatest number
Source:
1/24/2016
Determine the greatest number m such that the system
x
2
x^2
x
2
+
y
2
y^2
y
2
= 1; |
x
3
x^3
x
3
-
y
3
y^3
y
3
|+|x-y|=
m
3
m^3
m
3
has a solution.