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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2012 Serbia Team Selection Test
2012 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
3
1
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Isogonal conjugates, MP perpendicular to CF
Let
P
P
P
and
Q
Q
Q
be points inside triangle
A
B
C
ABC
A
BC
satisfying
∠
P
A
C
=
∠
Q
A
B
\angle PAC=\angle QAB
∠
P
A
C
=
∠
Q
A
B
and
∠
P
B
C
=
∠
Q
B
A
\angle PBC=\angle QBA
∠
PBC
=
∠
QB
A
.a) Prove that feet of perpendiculars from
P
P
P
and
Q
Q
Q
on the sides of triangle
A
B
C
ABC
A
BC
are concyclic. b) Let
D
D
D
and
E
E
E
be feet of perpendiculars from
P
P
P
on the lines
B
C
BC
BC
and
A
C
AC
A
C
and
F
F
F
foot of perpendicular from
Q
Q
Q
on
A
B
AB
A
B
. Let
M
M
M
be intersection point of
D
E
DE
D
E
and
A
B
AB
A
B
. Prove that
M
P
⊥
C
F
MP\bot CF
MP
⊥
CF
.
2
1
Hide problems
f(n)={m : m<=n, sigma(m) is odd}, f(n)|n, for inf. many n
Let
σ
(
x
)
\sigma(x)
σ
(
x
)
denote the sum of divisors of natural number
x
x
x
, including
1
1
1
and
x
x
x
. For every
n
∈
N
n\in \mathbb{N}
n
∈
N
define
f
(
n
)
f(n)
f
(
n
)
as number of natural numbers
m
,
m
≤
n
m, m\leq n
m
,
m
≤
n
, for which
σ
(
m
)
\sigma(m)
σ
(
m
)
is odd number. Prove that there are infinitely many natural numbers
n
n
n
, such that
f
(
n
)
∣
n
f(n)|n
f
(
n
)
∣
n
.
1
1
Hide problems
Polynomial satisfying P(a)^3+P(b)^3+P(c)^3 >= 3P(a)P(b)P(c)
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial of degree
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with real coefficients satisfying the condition
P
(
a
)
3
+
P
(
b
)
3
+
P
(
c
)
3
≥
3
P
(
a
)
P
(
b
)
P
(
c
)
,
P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),
P
(
a
)
3
+
P
(
b
)
3
+
P
(
c
)
3
≥
3
P
(
a
)
P
(
b
)
P
(
c
)
,
for all real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
. Is it possible for
P
(
x
)
P(x)
P
(
x
)
to have exactly
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distinct real roots?