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IFYM Int. Fest. of Young Mathematicians, Sozopol
2015 IFYM, Sozopol
2015 IFYM, Sozopol
Part of
IFYM Int. Fest. of Young Mathematicians, Sozopol
Subcontests
(8)
2
1
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Problem 2 of Third round
Let
A
B
C
D
ABCD
A
BC
D
be an inscribed quadrilateral and
P
P
P
be an inner point for it so that
∠
P
A
B
=
∠
P
B
C
=
∠
P
C
D
=
∠
P
D
A
\angle PAB=\angle PBC=\angle PCD=\angle PDA
∠
P
A
B
=
∠
PBC
=
∠
PC
D
=
∠
P
D
A
. The lines
A
D
AD
A
D
and
B
C
BC
BC
intersect in point
Q
Q
Q
and lines
A
B
AB
A
B
and
C
D
CD
C
D
– in point
R
R
R
. Prove that
∠
(
P
Q
,
P
R
)
=
∠
(
A
C
,
B
D
)
\angle (PQ,PR)=\angle (AC,BD)
∠
(
PQ
,
PR
)
=
∠
(
A
C
,
B
D
)
.
6
5
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4
3
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Problem 4 of First round
In how many ways can
n
n
n
rooks be placed on a
2
n
2n
2
n
x
2
n
2n
2
n
chessboard, so that they cover all the white fields?
Problem 4 of Second round - Injective function with perfect squares property
Let
k
k
k
be a natural number. For each natural number
n
n
n
we define
f
k
(
n
)
f_k (n)
f
k
(
n
)
to be the least number, greater than
k
n
kn
kn
, for which
n
f
k
(
n
)
nf_k (n)
n
f
k
(
n
)
is a perfect square. Prove that
f
k
(
n
)
f_k (n)
f
k
(
n
)
is injective.
Problem 4 of Finals
For all real numbers
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
such that
a
b
c
=
1
abc=1
ab
c
=
1
, prove that
a
1
+
b
3
+
b
1
+
c
3
+
c
1
+
a
3
≥
3
2
\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}
1
+
b
3
a
+
1
+
c
3
b
+
1
+
a
3
c
≥
2
3
.
8
5
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5
5
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7
5
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1
5
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3
4
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