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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2019 Poland - Second Round
2019 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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Point in the interior
Let
X
X
X
be a point lying in the interior of the acute triangle
A
B
C
ABC
A
BC
such that \begin{align*} \sphericalangle BAX = 2\sphericalangle XBA \ \ \ \ \hbox{and} \ \ \ \ \sphericalangle XAC = 2\sphericalangle ACX. \end{align*} Denote by
M
M
M
the midpoint of the arc
B
C
BC
BC
of the circumcircle
(
A
B
C
)
(ABC)
(
A
BC
)
containing
A
A
A
. Prove that
X
M
=
X
A
XM=XA
XM
=
X
A
.
5
1
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Sequence of integers
Let
b
0
,
b
1
,
b
2
,
…
b_0, b_1, b_2, \ldots
b
0
,
b
1
,
b
2
,
…
be a sequence of pairwise distinct nonnegative integers such that
b
0
=
0
b_0=0
b
0
=
0
and
b
n
<
2
n
b_n<2n
b
n
<
2
n
for all positive integers
n
n
n
. Prove that for each nonnegative integer
m
m
m
there exist nonnegative integers
k
,
ℓ
k, \ell
k
,
ℓ
such that \begin{align*} b_k+b_{\ell}=m. \end{align*}
4
1
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Coprime integers divisibility
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
(
n
≥
3
n\ge 3
n
≥
3
) be positive integers such that
g
c
d
(
a
1
,
a
2
,
…
,
a
n
)
=
1
gcd(a_1, a_2, \ldots, a_n)=1
g
c
d
(
a
1
,
a
2
,
…
,
a
n
)
=
1
and for each
i
∈
{
1
,
2
,
…
,
n
}
i\in \lbrace 1,2,\ldots, n \rbrace
i
∈
{
1
,
2
,
…
,
n
}
we have
a
i
∣
a
1
+
a
2
+
…
+
a
n
a_i|a_1+a_2+\ldots+a_n
a
i
∣
a
1
+
a
2
+
…
+
a
n
. Prove that
a
1
a
2
…
a
n
∣
(
a
1
+
a
2
+
…
+
a
n
)
n
−
2
a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}
a
1
a
2
…
a
n
∣
(
a
1
+
a
2
+
…
+
a
n
)
n
−
2
.
3
1
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Functions and rational numbers
Let
f
(
t
)
=
t
3
+
t
f(t)=t^3+t
f
(
t
)
=
t
3
+
t
. Decide if there exist rational numbers
x
,
y
x, y
x
,
y
and positive integers
m
,
n
m, n
m
,
n
such that
x
y
=
3
xy=3
x
y
=
3
and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}
2
1
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Determine all integers
Determine all nonnegative integers
x
,
y
x, y
x
,
y
satisfying the equation \begin{align*} \sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}. \end{align*}
1
1
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Lengths of arcs
A cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
is given. Point
K
1
,
K
2
K_1, K_2
K
1
,
K
2
lie on the segment
A
B
AB
A
B
, points
L
1
,
L
2
L_1, L_2
L
1
,
L
2
on the segment
B
C
BC
BC
, points
M
1
,
M
2
M_1, M_2
M
1
,
M
2
on the segment
C
D
CD
C
D
and points
N
1
,
N
2
N_1, N_2
N
1
,
N
2
on the segment
D
A
DA
D
A
. Moreover, points
K
1
,
K
2
,
L
1
,
L
2
,
M
1
,
M
2
,
N
1
,
N
2
K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2
K
1
,
K
2
,
L
1
,
L
2
,
M
1
,
M
2
,
N
1
,
N
2
lie on a circle
ω
\omega
ω
in that order. Denote by
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
the lengths of the arcs
N
2
K
1
,
K
2
L
1
,
L
2
M
1
,
M
2
N
1
N_2K_1, K_2L_1, L_2M_1, M _2N_1
N
2
K
1
,
K
2
L
1
,
L
2
M
1
,
M
2
N
1
of the circle
ω
\omega
ω
not containing points
K
2
,
L
2
,
M
2
,
N
2
K_2, L_2, M_2, N_2
K
2
,
L
2
,
M
2
,
N
2
, respectively. Prove that \begin{align*} a+c=b+d. \end{align*}