MathDB
Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2006 Hong kong National Olympiad
2006 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(4)
4
1
Hide problems
The constants M and M'
Let
(
a
n
)
n
≥
1
(a_n)_{n\ge 1}
(
a
n
)
n
≥
1
be a sequence of positive numbers. If there is a constant
M
>
0
M > 0
M
>
0
such that
a
2
2
+
a
2
2
+
…
+
a
n
2
<
M
a
n
+
1
2
a_2^2 + a_2^2 +\ldots + a_n^2 < Ma_{n+1}^2
a
2
2
+
a
2
2
+
…
+
a
n
2
<
M
a
n
+
1
2
for all
n
n
n
, then prove that there is a constant
M
′
>
0
M ' > 0
M
′
>
0
such that
a
1
+
a
2
+
…
+
a
n
<
M
′
a
n
+
1
a_1 + a_2 +\ldots + a_n < M ' a_{n+1}
a
1
+
a
2
+
…
+
a
n
<
M
′
a
n
+
1
.
2
1
Hide problems
The square of the sum of the digits of k function
For a positive integer
k
k
k
, let
f
1
(
k
)
f_1(k)
f
1
(
k
)
be the square of the sum of the digits of
k
k
k
. Define
f
n
+
1
f_{n+1}
f
n
+
1
=
f
1
∘
f
n
f_1 \circ f_n
f
1
∘
f
n
. Evaluate
f
2007
(
2
2006
)
f_{2007}(2^{2006} )
f
2007
(
2
2006
)
.
1
1
Hide problems
Any three elements of a subset
A subset
M
M
M
of
{
1
,
2
,
.
.
.
,
2006
}
\{1, 2, . . . , 2006\}
{
1
,
2
,
...
,
2006
}
has the property that for any three elements
x
,
y
,
z
x, y, z
x
,
y
,
z
of
M
M
M
with
x
<
y
<
z
x < y < z
x
<
y
<
z
,
x
+
y
x+ y
x
+
y
does not divide
z
z
z
. Determine the largest possible size of
M
M
M
.
3
1
Hide problems
Collinearity in cyclic quadrilateral
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
C
≠
B
D
AC \neq BD
A
C
=
B
D
is inscribed in a circle with center
O
O
O
. Let
E
E
E
be the intersection of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. If
P
P
P
is a point inside
A
B
C
D
ABCD
A
BC
D
such that
∠
P
A
B
+
∠
P
C
B
=
∠
P
B
C
+
∠
P
D
C
=
9
0
∘
\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ
∠
P
A
B
+
∠
PCB
=
∠
PBC
+
∠
P
D
C
=
9
0
∘
, prove that
O
O
O
,
P
P
P
and
E
E
E
are collinear.