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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 11th XMO
the 11th XMO
Part of
XES Mathematics Olympiad
Subcontests
(6)
4
1
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BeeHive!!!
We define a beehive of order
n
n
n
as follows:a beehive of order 1 is one hexagonTo construct a beehive of order
n
n
n
, take a beehive of order
n
−
1
n-1
n
−
1
and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of
n
=
2
,
3
n=2,3
n
=
2
,
3
Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.Let
f
(
n
)
f(n)
f
(
n
)
be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find
f
(
n
)
f(n)
f
(
n
)
.
3
1
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Another standard NT Cookie
Let
p
p
p
is a prime and
p
≡
2
(
m
o
d
3
)
p\equiv 2\pmod 3
p
≡
2
(
mod
3
)
. For
∀
a
∈
Z
\forall a\in\mathbb Z
∀
a
∈
Z
, if
p
∣
∏
i
=
1
p
(
i
3
−
a
i
−
1
)
,
p\mid \prod\limits_{i=1}^p(i^3-ai-1),
p
∣
i
=
1
∏
p
(
i
3
−
ai
−
1
)
,
then
a
a
a
is called a "GuGu" number. How many "GuGu" numbers are there in the set
{
1
,
2
,
⋯
,
p
}
?
\{1,2,\cdots ,p\}?
{
1
,
2
,
⋯
,
p
}?
(We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)
1
1
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Geometry in XMO, don't discuss until tomorrow
Let
△
A
B
C
\triangle ABC
△
A
BC
be connected to the circle
Γ
\Gamma
Γ
. The angular bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects
B
C
BC
BC
to
D
D
D
. Straight line
B
P
BP
BP
intersects
A
C
AC
A
C
to
E
E
E
, and straight line
C
P
CP
CP
intersects
A
B
AB
A
B
to
F
F
F
. Let the tangent of the circle
Γ
\Gamma
Γ
at point
A
A
A
intersect the line
E
F
EF
EF
at the point
Q
Q
Q
. Proof:
P
Q
∥
B
C
PQ\parallel BC
PQ
∥
BC
.
10
1
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complex number problem, don't discuss until Tuesday
Given
t
∈
C
t\in\mathbb C
t
∈
C
. Complex numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy that
∣
x
∣
=
∣
y
∣
=
∣
z
∣
=
1
|x|=|y|=|z|=1
∣
x
∣
=
∣
y
∣
=
∣
z
∣
=
1
and
t
y
=
1
x
+
1
z
\frac{t}{y}=\frac{1}{x}+\frac{1}{z}
y
t
=
x
1
+
z
1
. Calculate
∣
2
x
y
+
2
y
z
+
3
z
x
x
+
y
+
z
∣
.
\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.
x
+
y
+
z
2
x
y
+
2
yz
+
3
z
x
.
2
1
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Hard 3 variable inequality
Suppose
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
and
a
b
c
=
64
abc=64
ab
c
=
64
, show that
∑
c
y
c
a
2
a
3
+
8
b
3
+
8
≥
2
3
\sum_{cyc}\frac{a^2}{\sqrt{a^3+8}\sqrt{b^3+8}}\ge\frac{2}{3}
cyc
∑
a
3
+
8
b
3
+
8
a
2
≥
3
2
9
1
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two variable inequality
x
,
y
∈
R
,
(
4
x
3
−
3
x
)
2
+
(
4
y
3
−
3
y
)
2
=
1.
Find the maximum of
x
+
y
.
x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.
x
,
y
∈
R
,
(
4
x
3
−
3
x
)
2
+
(
4
y
3
−
3
y
)
2
=
1.
Find the maximum of
x
+
y
.