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Contests
International Contests
Silk Road
2014 Silk Road
2014 Silk Road
Part of
Silk Road
Subcontests
(4)
2
1
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SRMC 2014
Let
w
w
w
be the circumcircle of non-isosceles acute triangle
A
B
C
ABC
A
BC
. Tangent lines to
w
w
w
in
A
A
A
and
B
B
B
intersect at point
S
S
S
. Let M be the midpoint of
A
B
AB
A
B
, and
H
H
H
be the orthocenter of triangle
A
B
C
ABC
A
BC
. The line
H
A
HA
H
A
intersects lines
C
M
CM
CM
and
C
S
CS
CS
at points
M
a
M_a
M
a
and
S
a
S_a
S
a
, respectively. The points
M
b
M_b
M
b
and
S
b
S_b
S
b
are defined analogously. Prove that
M
a
S
b
M_aS_b
M
a
S
b
and
M
b
S
a
M_bS_a
M
b
S
a
are the altitudes of triangle
M
a
M
b
H
M_aM_bH
M
a
M
b
H
.
1
1
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SRMC 2014 problem 1
What is the maximum number of coins can be arranged in cells of the table
n
×
n
n \times n
n
×
n
(each cell is not more of the one coin) so that any coin was not simultaneously below and to the right than any other?
3
1
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Inequality from SRMC
a
,
b
,
c
≥
0
,
a
3
+
b
3
+
c
3
+
a
b
c
=
4
a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4
a
,
b
,
c
≥
0
,
a
3
+
b
3
+
c
3
+
ab
c
=
4
Prove that
a
3
b
+
b
3
c
+
c
3
b
≤
3
a^3b+b^3c+c^3b \le 3
a
3
b
+
b
3
c
+
c
3
b
≤
3
4
1
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Nice and Hard FE
Find all
f
:
N
→
N
f:N\rightarrow N
f
:
N
→
N
, such that
∀
m
,
n
∈
N
\forall m,n\in N
∀
m
,
n
∈
N
2
f
(
m
n
)
≥
f
(
m
2
+
n
2
)
−
f
(
m
)
2
−
f
(
n
)
2
≥
2
f
(
m
)
f
(
n
)
2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n)
2
f
(
mn
)
≥
f
(
m
2
+
n
2
)
−
f
(
m
)
2
−
f
(
n
)
2
≥
2
f
(
m
)
f
(
n
)