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Problems
Contests
International Contests
Silk Road
2004 Silk Road
2004 Silk Road
Part of
Silk Road
Subcontests
(4)
4
1
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International competition SRMC 2004 P-4
Natural
n
≥
2
n \geq 2
n
≥
2
is given. Group of people calls
n
−
c
o
m
p
a
c
t
n-compact
n
−
co
m
p
a
c
t
, if for any men from group, we can found
n
n
n
people (without he), each two of there are familiar. Find maximum
N
N
N
such that for any
n
−
c
o
m
p
a
c
t
n-compact
n
−
co
m
p
a
c
t
group, consisting
N
N
N
people contains subgroup from
n
+
1
n+1
n
+
1
people, each of two of there are familiar.
3
1
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International competition SRMC 2004 P-3
In-circle of
A
B
C
ABC
A
BC
with center
I
I
I
touch
A
B
AB
A
B
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
respectively.
B
I
BI
B
I
and
C
I
CI
C
I
intersect
P
Q
PQ
PQ
at
K
K
K
and
L
L
L
respectively. Prove, that circumcircle of
I
L
K
ILK
I
L
K
touch incircle of
A
B
C
ABC
A
BC
iff
∣
A
B
∣
+
∣
A
C
∣
=
3
∣
B
C
∣
|AB|+|AC|=3|BC|
∣
A
B
∣
+
∣
A
C
∣
=
3∣
BC
∣
.
2
1
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International competition SRMC 2004 P-2
find all primes
p
p
p
, for which exist natural numbers, such that
p
=
m
2
+
n
2
p=m^2+n^2
p
=
m
2
+
n
2
and
p
∣
(
m
3
+
n
3
−
4
)
p|(m^3+n^3-4)
p
∣
(
m
3
+
n
3
−
4
)
.
1
1
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International competition SRMC 2004 P-1
Find all
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
, such that
(
x
+
y
)
(
f
(
x
)
−
f
(
y
)
)
=
(
x
−
y
)
f
(
x
+
y
)
(x+y)(f(x)-f(y))=(x-y)f(x+y)
(
x
+
y
)
(
f
(
x
)
−
f
(
y
))
=
(
x
−
y
)
f
(
x
+
y
)
for all real
x
,
y
x,y
x
,
y
.