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National and Regional Contests
Poland Contests
Poland - Second Round
2022 Poland - Second Round
2022 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
3
1
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Polish MO 2022 P3
Positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying the equation
a
3
+
4
b
+
c
=
a
b
c
,
a^3+4b+c = abc,
a
3
+
4
b
+
c
=
ab
c
,
where
a
≥
c
a \geq c
a
≥
c
and the number
p
=
a
2
+
2
a
+
2
p = a^2+2a+2
p
=
a
2
+
2
a
+
2
is a prime. Prove that
p
p
p
divides
a
+
2
b
+
2
a+2b+2
a
+
2
b
+
2
.
2
1
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Polish MO 2022 P2
Given a cyclic quadriteral
A
B
C
D
ABCD
A
BC
D
. The circumcenter lies in the quadriteral
A
B
C
D
ABCD
A
BC
D
. Diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersects at
S
S
S
. Points
P
P
P
and
Q
Q
Q
are the midpoints of
A
D
AD
A
D
and
B
C
BC
BC
. Let
p
p
p
be a line perpendicular to
A
C
AC
A
C
through
P
P
P
,
q
q
q
perpendicular line to
B
D
BD
B
D
through
Q
Q
Q
and
s
s
s
perpendicular to
C
D
CD
C
D
through
S
S
S
. Prove that
p
,
q
,
s
p,q,s
p
,
q
,
s
intersects at one point.
1
1
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Polish MO 2022 P1
Find all real quadruples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
satisfying the system of equations
{
a
b
+
c
d
=
6
a
c
+
b
d
=
3
a
d
+
b
c
=
2
a
+
b
+
c
+
d
=
6.
\left\{ \begin{array}{ll} ab+cd = 6 \\ ac + bd = 3 \\ ad + bc = 2 \\ a + b + c + d = 6. \end{array} \right.
⎩
⎨
⎧
ab
+
c
d
=
6
a
c
+
b
d
=
3
a
d
+
b
c
=
2
a
+
b
+
c
+
d
=
6.
6
1
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Polish MO 2022 P6
n
n
n
players took part in badminton tournament, where
n
n
n
is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.
5
1
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Polish MO 2022 P5
Let
n
n
n
be an positive integer. We call
n
n
n
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
when there exists positive integer
k
k
k
s.t.
n
=
k
(
k
+
1
)
n=k(k+1)
n
=
k
(
k
+
1
)
. Does there exist 2022 pairwise distinct
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
numbers s.t. their sum is also
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
number?
4
1
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Polish MO 2022 P4
Given quadrilateral
A
B
C
D
ABCD
A
BC
D
inscribed into a circle with diagonal
A
C
AC
A
C
as diameter. Let
E
E
E
be a point on segment
B
C
BC
BC
s.t.
∢
D
A
C
=
∢
E
A
B
\sphericalangle DAC=\sphericalangle EAB
∢
D
A
C
=
∢
E
A
B
. Point
M
M
M
is midpoint of
C
E
CE
CE
. Prove that
B
M
=
D
M
BM=DM
BM
=
D
M
.