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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2020 Poland - Second Round
2020 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6.
1
Hide problems
Amazing Algebra Problem
Let
(
a
0
,
a
1
,
a
2
,
.
.
.
)
(a_0,a_1,a_2,...)
(
a
0
,
a
1
,
a
2
,
...
)
and
(
b
0
,
b
1
,
b
2
,
.
.
.
)
(b_0,b_1,b_2,...)
(
b
0
,
b
1
,
b
2
,
...
)
be such sequences of non-negative real numbers, that for every integer
i
⩾
1
i\geqslant 1
i
⩾
1
holds
a
i
2
⩽
a
i
−
1
a
i
+
1
a_i^2\leqslant a_{i-1}a_{i+1}
a
i
2
⩽
a
i
−
1
a
i
+
1
and
b
i
2
⩽
b
i
−
1
b
i
+
1
b_i^2\leqslant b_{i-1}b_{i+1}
b
i
2
⩽
b
i
−
1
b
i
+
1
. Define sequence
c
0
,
c
1
,
c
2
,
.
.
.
c_0,c_1,c_2,...
c
0
,
c
1
,
c
2
,
...
as
c
0
=
a
0
b
0
,
c
n
=
∑
i
=
0
n
(
n
i
)
a
i
b
n
−
i
.
c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.
c
0
=
a
0
b
0
,
c
n
=
i
=
0
∑
n
(
i
n
)
a
i
b
n
−
i
.
Prove that for every integer
k
⩾
1
k\geqslant 1
k
⩾
1
holds
c
k
2
⩽
c
k
−
1
c
k
+
1
c_{k}^2\leqslant c_{k-1}c_{k+1}
c
k
2
⩽
c
k
−
1
c
k
+
1
.
5.
1
Hide problems
Number Theory
Let
p
>
p>
p
>
be a prime number and
S
S
S
be a set of
p
+
1
p+1
p
+
1
integers. Prove that there exist pairwise distinct numbers
a
1
,
a
2
,
.
.
.
,
a
p
−
1
∈
S
a_1,a_2,...,a_{p-1}\in S
a
1
,
a
2
,
...
,
a
p
−
1
∈
S
that
a
1
+
2
a
2
+
3
a
3
+
.
.
.
+
(
p
−
1
)
a
p
−
1
a_1+2a_2+3a_3+...+(p-1)a_{p-1}
a
1
+
2
a
2
+
3
a
3
+
...
+
(
p
−
1
)
a
p
−
1
is divisible by
p
p
p
.
4.
1
Hide problems
Easy Geometry - Hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a such convex hexagon that
A
B
=
C
D
=
E
F
and
B
C
=
D
E
=
.
F
A
AB=CD=EF\; \text{and} \; BC=DE=.FA
A
B
=
C
D
=
EF
and
BC
=
D
E
=
.
F
A
Prove that if
∢
F
A
B
+
∢
A
B
C
=
∢
F
A
B
+
∢
E
F
A
=
24
0
∘
\sphericalangle FAB + \sphericalangle ABC=\sphericalangle FAB + \sphericalangle EFA = 240^{\circ}
∢
F
A
B
+
∢
A
BC
=
∢
F
A
B
+
∢
EF
A
=
24
0
∘
, then
∢
F
A
B
+
∢
C
D
E
=
24
0
∘
\sphericalangle FAB+\sphericalangle CDE=240^{\circ}
∢
F
A
B
+
∢
C
D
E
=
24
0
∘
.
3.
1
Hide problems
Geometry
Let
M
M
M
be the midpoint of the side
B
C
BC
BC
of a acute triangle
A
B
C
ABC
A
BC
. Incircle of the triangle
A
B
M
ABM
A
BM
is tangent to the side
A
B
AB
A
B
at the point
D
D
D
. Incircle of the triangle
A
C
M
ACM
A
CM
is tangent to the side
A
C
AC
A
C
at the point
E
E
E
. Let
F
F
F
be the such point, that the quadrilateral
D
M
E
F
DMEF
D
MEF
is a parallelogram. Prove that
F
F
F
lies on the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
2.
1
Hide problems
Polish Combinatorics
Let
n
n
n
be a positive integer. Jadzia has to write all integers from
1
1
1
to
2
n
−
1
2n-1
2
n
−
1
on a board, and she writes each integer in blue or red color. We say that pair of numbers
i
,
j
∈
{
1
,
2
,
3
,
.
.
.
,
2
n
−
1
}
i,j\in \{1,2,3,...,2n-1\}
i
,
j
∈
{
1
,
2
,
3
,
...
,
2
n
−
1
}
, where
i
⩽
j
i\leqslant j
i
⩽
j
, is
<
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
>
if and only if number of blue numbers among
i
,
i
+
1
,
.
.
.
,
j
i,i+1,...,j
i
,
i
+
1
,
...
,
j
is odd. Determine, in terms of
n
n
n
, maximal number of good pairs.
1.
1
Hide problems
Algebra problem
Assume that for pairwise distinct real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
holds:
(
a
2
+
b
2
−
1
)
(
a
+
b
)
=
(
b
2
+
c
2
−
1
)
(
b
+
c
)
=
(
c
2
+
d
2
−
1
)
(
c
+
d
)
.
(a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d).
(
a
2
+
b
2
−
1
)
(
a
+
b
)
=
(
b
2
+
c
2
−
1
)
(
b
+
c
)
=
(
c
2
+
d
2
−
1
)
(
c
+
d
)
.
Prove that
a
+
b
+
c
+
d
=
0.
a+b+c+d=0.
a
+
b
+
c
+
d
=
0.