MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
1996 China National Olympiad
1996 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
Functional Equation involving cubes
Suppose that the function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
satisfies
f
(
x
3
+
y
3
)
=
(
x
+
y
)
(
f
(
x
)
2
−
f
(
x
)
f
(
y
)
+
f
(
y
)
2
)
f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)
f
(
x
3
+
y
3
)
=
(
x
+
y
)
(
f
(
x
)
2
−
f
(
x
)
f
(
y
)
+
f
(
y
)
2
)
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
. Prove that
f
(
1996
x
)
=
1996
f
(
x
)
f(1996x)=1996f(x)
f
(
1996
x
)
=
1996
f
(
x
)
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
.
Minimum value of all the longest side
In the triangle
A
B
C
ABC
A
BC
,
∠
C
=
9
0
∘
,
∠
A
=
3
0
∘
\angle{C}=90^{\circ},\angle {A}=30^{\circ}
∠
C
=
9
0
∘
,
∠
A
=
3
0
∘
and
B
C
=
1
BC=1
BC
=
1
. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle
A
B
C
ABC
A
BC
.
2
2
Hide problems
Sum of x is 1
Let
n
n
n
be a natural number. Suppose that
x
0
=
0
x_0=0
x
0
=
0
and that
x
i
>
0
x_i>0
x
i
>
0
for all
i
∈
{
1
,
2
,
…
,
n
}
i\in\{1,2,\ldots ,n\}
i
∈
{
1
,
2
,
…
,
n
}
. If
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^nx_i=1
∑
i
=
1
n
x
i
=
1
, prove that
1
≤
∑
i
=
1
n
x
i
1
+
x
0
+
x
1
+
…
+
x
i
−
1
x
i
+
…
+
x
n
<
π
2
1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2}
1
≤
i
=
1
∑
n
1
+
x
0
+
x
1
+
…
+
x
i
−
1
x
i
+
…
+
x
n
x
i
<
2
π
a+b|ab
Find the smallest positive integer
K
K
K
such that every
K
K
K
-element subset of
{
1
,
2
,
.
.
.
,
50
}
\{1,2,...,50 \}
{
1
,
2
,
...
,
50
}
contains two distinct elements
a
,
b
a,b
a
,
b
such that a\plus{}b divides
a
b
ab
ab
.
1
2
Hide problems
Orthocentre is collinear with two tangent points
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be a triangle with orthocentre
H
H
H
. The tangent lines from
A
A
A
to the circle with diameter
B
C
BC
BC
touch this circle at
P
P
P
and
Q
Q
Q
. Prove that
H
,
P
H,P
H
,
P
and
Q
Q
Q
are collinear.
Singers and concerts
8
8
8
singers take part in a festival. The organiser wants to plan
m
m
m
concerts. For every concert there are
4
4
4
singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises
m
m
m
.