MathDB
Problems
Contests
National and Regional Contests
Hong Kong Contests
Hong Kong National Olympiad
2001 Hong kong National Olympiad
2001 Hong kong National Olympiad
Part of
Hong Kong National Olympiad
Subcontests
(4)
1
1
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smaller arcs are equal
A triangle
A
B
C
ABC
A
BC
is given. A circle
Γ
\Gamma
Γ
, passing through
A
A
A
, is tangent to side
B
C
BC
BC
at point
P
P
P
and intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
at
M
M
M
and
N
N
N
respectively. Prove that the smaller arcs
M
P
MP
MP
and
N
P
NP
NP
of
Γ
\Gamma
Γ
are equal iff
Γ
\Gamma
Γ
is tangent to the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
at
A
A
A
.
2
1
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integer equation
Find, with proof, all positive integers
n
n
n
such that the equation
x
3
+
y
3
+
z
3
=
n
x
2
y
2
z
2
x^{3}+y^{3}+z^{3}=nx^{2}y^{2}z^{2}
x
3
+
y
3
+
z
3
=
n
x
2
y
2
z
2
has a solution in positive integers.
4
1
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$212$ points inside or on a given unit circle
There are
212
212
212
points inside or on a given unit circle. Prove that there are at least
2001
2001
2001
pairs of points having distances at most
1
1
1
.
3
1
Hide problems
$P(x)\in\mathbb{Z}[x]$
Let
k
≥
4
k\geq 4
k
≥
4
be an integer number.
P
(
x
)
∈
Z
[
x
]
P(x)\in\mathbb{Z}[x]
P
(
x
)
∈
Z
[
x
]
such that
0
≤
P
(
c
)
≤
k
0\leq P(c)\leq k
0
≤
P
(
c
)
≤
k
for all
c
=
0
,
1
,
.
.
.
,
k
+
1
c=0,1,...,k+1
c
=
0
,
1
,
...
,
k
+
1
. Prove that
P
(
0
)
=
P
(
1
)
=
.
.
.
=
P
(
k
+
1
)
P(0)=P(1)=...=P(k+1)
P
(
0
)
=
P
(
1
)
=
...
=
P
(
k
+
1
)
.