MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2018 Poland - Second Round
2018 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
Sequence of roots
Let
k
k
k
be a positive integer and
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
be a sequence of terms from set
{
0
,
1
,
.
.
.
,
k
}
\{ 0, 1, ..., k \}
{
0
,
1
,
...
,
k
}
. Let
b
n
=
a
1
n
+
a
2
n
+
.
.
.
+
a
n
n
n
b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}
b
n
=
n
a
1
n
+
a
2
n
+
...
+
a
n
n
for all positive integers
n
n
n
. Prove, that if in sequence
b
1
,
b
2
,
b
3
,
.
.
.
b_1, b_2, b_3, ...
b
1
,
b
2
,
b
3
,
...
are infinitely many integers, then all terms of this series are integers.
5
1
Hide problems
5-element subsets
Let
A
1
,
A
2
,
.
.
.
,
A
k
A_1, A_2, ..., A_k
A
1
,
A
2
,
...
,
A
k
be
5
5
5
-element subsets of set
{
1
,
2
,
.
.
.
,
23
}
\{1, 2, ..., 23\}
{
1
,
2
,
...
,
23
}
such that, for all
1
≤
i
<
j
≤
k
1 \le i < j \le k
1
≤
i
<
j
≤
k
set
A
i
∩
A
j
A_i \cap A_j
A
i
∩
A
j
has at most three elements. Show that
k
≤
2018
k \le 2018
k
≤
2018
.
4
1
Hide problems
Trapezoid and tangent circles
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with bases
A
B
AB
A
B
and
C
D
CD
C
D
. Circle of diameter
B
C
BC
BC
is tangent to line
A
D
AD
A
D
. Prove, that circle of diameter
A
D
AD
A
D
is tangent to line
B
C
BC
BC
.
3
1
Hide problems
Bisector, orthogonal projection
Bisector of side
B
C
BC
BC
intersects circumcircle of triangle
A
B
C
ABC
A
BC
in points
P
P
P
and
Q
Q
Q
. Points
A
A
A
and
P
P
P
lie on the same side of line
B
C
BC
BC
. Point
R
R
R
is an orthogonal projection of point
P
P
P
on line
A
C
AC
A
C
. Point
S
S
S
is middle of line segment
A
Q
AQ
A
Q
. Show that points
A
,
B
,
R
,
S
A, B, R, S
A
,
B
,
R
,
S
lie on one circle.
2
1
Hide problems
Divisors inequality
Let
n
n
n
be a positive integer, which gives remainder
4
4
4
of dividing by
8
8
8
. Numbers
1
=
k
1
<
k
2
<
.
.
.
<
k
m
=
n
1 = k_1 < k_2 < ... < k_m = n
1
=
k
1
<
k
2
<
...
<
k
m
=
n
are all positive diivisors of
n
n
n
. Show that if
i
∈
{
1
,
2
,
.
.
.
,
m
−
1
}
i \in \{ 1, 2, ..., m - 1 \}
i
∈
{
1
,
2
,
...
,
m
−
1
}
isn't divisible by
3
3
3
, then
k
i
+
1
≤
2
k
i
k_{i + 1} \le 2k_{i}
k
i
+
1
≤
2
k
i
.
1
1
Hide problems
Determine all functions
Determine all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
which satisfy conditions:
f
(
x
)
+
f
(
y
)
≥
x
y
f(x) + f(y) \ge xy
f
(
x
)
+
f
(
y
)
≥
x
y
for all real
x
,
y
x, y
x
,
y
and for each real
x
x
x
exists real
y
y
y
, such that
f
(
x
)
+
f
(
y
)
=
x
y
f(x) + f(y) = xy
f
(
x
)
+
f
(
y
)
=
x
y
.