MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2021 Poland - Second Round
2021 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
f (a_1,..., a_p) - k$ is divisible by P, f (x_1,..., x_p)=x_1+2x_2+...+ px_p
Let
p
≥
5
p\ge 5
p
≥
5
be a prime number. Consider the function given by the formula
f
(
x
1
,
.
.
.
,
x
p
)
=
x
1
+
2
x
2
+
.
.
.
+
p
x
p
.
f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.
f
(
x
1
,
...
,
x
p
)
=
x
1
+
2
x
2
+
...
+
p
x
p
.
Let
A
k
A_k
A
k
denote the set of all these permutations
(
a
1
,
.
.
.
,
a
p
)
(a_1,..., a_p)
(
a
1
,
...
,
a
p
)
of the set
{
1
,
.
.
.
,
p
}
\{1,..., p\}
{
1
,
...
,
p
}
, for integer number
f
(
a
1
,
.
.
.
,
a
p
)
−
k
f (a_1,..., a_p) - k
f
(
a
1
,
...
,
a
p
)
−
k
is divisible by
p
p
p
and
a
i
≠
i
a_i \ne i
a
i
=
i
for all
i
∈
{
1
,
.
.
.
,
p
}
i \in \{1,..., p\}
i
∈
{
1
,
...
,
p
}
. Prove that the sets
A
1
A_1
A
1
and
A
4
A_4
A
4
have the same number of elements.
5
1
Hide problems
combo geo with disjoint rectangles, sides // to axis
Find the largest positive integer
n
n
n
with the following property: there are rectangles
A
1
,
.
.
.
,
A
n
A_1, ... , A_n
A
1
,
...
,
A
n
and
B
1
,
.
.
.
,
B
n
,
B_1,... , B_n,
B
1
,
...
,
B
n
,
on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles
A
i
A_i
A
i
and
B
i
B_i
B
i
are disjoint for all
i
∈
{
1
,
.
.
.
,
n
}
i \in \{1,..., n\}
i
∈
{
1
,
...
,
n
}
, but the rectangles
A
i
A_i
A
i
and
B
j
B_j
B
j
have a common point for all
i
,
j
∈
{
1
,
.
.
.
,
n
}
i, j \in \{1,..., n\}
i
,
j
∈
{
1
,
...
,
n
}
,
i
≠
j
i \ne j
i
=
j
.Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices
4
1
Hide problems
x^2+1/x^2+y^2+1/y^2 rational if x+1/x+y+1/y, x^3+1/x^3+y^3+1/y^3 rationals
There are real numbers
x
,
y
x, y
x
,
y
such that
x
≠
0
x \ne 0
x
=
0
,
y
≠
0
y \ne 0
y
=
0
,
x
y
+
1
≠
0
xy + 1 \ne 0
x
y
+
1
=
0
and
x
+
y
≠
0
x + y \ne 0
x
+
y
=
0
. Suppose the numbers
x
+
1
x
+
y
+
1
y
x + \frac{1}{x} + y + \frac{1}{y}
x
+
x
1
+
y
+
y
1
and
x
3
+
1
x
3
+
y
3
+
1
y
3
x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}
x
3
+
x
3
1
+
y
3
+
y
3
1
are rational. Prove that then the number
x
2
+
1
x
2
+
y
2
+
1
y
2
x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}
x
2
+
x
2
1
+
y
2
+
y
2
1
is also rational.
2
1
Hide problems
AO = OC wanted, <DBA = <CBP, # ABCD
The point P lies on the side
C
D
CD
C
D
of the parallelogram
A
B
C
D
ABCD
A
BC
D
with
∠
D
B
A
=
∠
C
B
P
\angle DBA = \angle CBP
∠
D
B
A
=
∠
CBP
. Point
O
O
O
is the center of the circle passing through the points
D
D
D
and
P
P
P
and tangent to the straight line
A
D
AD
A
D
at point
D
D
D
. Prove that
A
O
=
O
C
AO = OC
A
O
=
OC
.
1
1
Hide problems
coloring n numbered cards from 1 to n with 3 colors
Jacek has
n
n
n
cards numbered consecutively with the numbers
1
,
.
.
.
,
n
1,. . . , n
1
,
...
,
n
, which he places in a row on the table, in any order he chooses. Jacek will remove cards from the table in the sequence consistent with the numbering of cards: first they will remove the card number
1
1
1
, then the card number
2
2
2
, and so on. Before Jacek starts taking the cards, Pie will color each one of cards in red, blue or yellow. Prove that Pie can color the cards in such a way that when Jacek takes them off, it will be fulfilled at every moment the following condition: between any two cards of the same suit there is at least one card of a different color.
3
1
Hide problems
Number Theory
Positive integers
a
,
b
,
z
a,b,z
a
,
b
,
z
satisfy the equation
a
b
=
z
2
+
1
ab=z^2+1
ab
=
z
2
+
1
. Prove that there exist positive integers
x
,
y
x,y
x
,
y
such that
a
b
=
x
2
+
1
y
2
+
1
\frac{a}{b}=\frac{x^2+1}{y^2+1}
b
a
=
y
2
+
1
x
2
+
1