MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2006 Poland - Second Round
2006 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
2
1
Hide problems
Perpendicular lines plus two circles
Point
C
C
C
is a midpoint of
A
B
AB
A
B
. Circle
o
1
o_1
o
1
which passes through
A
A
A
and
C
C
C
intersect circle
o
2
o_2
o
2
which passes through
B
B
B
and
C
C
C
in two different points
C
C
C
and
D
D
D
. Point
P
P
P
is a midpoint of arc
A
D
AD
A
D
of circle
o
1
o_1
o
1
which doesn't contain
C
C
C
. Point
Q
Q
Q
is a midpoint of arc
B
D
BD
B
D
of circle
o
2
o_2
o
2
which doesn't contain
C
C
C
. Prove that
P
Q
⊥
C
D
PQ \perp CD
PQ
⊥
C
D
.
3
2
Hide problems
classical - symmetrical with 3 variables
Positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
b
+
b
c
+
c
a
=
a
b
c
ab+bc+ca=abc
ab
+
b
c
+
c
a
=
ab
c
. Prove that:
a
4
+
b
4
a
b
(
a
3
+
b
3
)
+
b
4
+
c
4
b
c
(
b
3
+
c
3
)
+
c
4
+
a
4
c
a
(
c
3
+
a
3
)
≥
1
\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1
ab
(
a
3
+
b
3
)
a
4
+
b
4
+
b
c
(
b
3
+
c
3
)
b
4
+
c
4
+
c
a
(
c
3
+
a
3
)
c
4
+
a
4
≥
1
largest possible cardinality of a set
Given is a prime number
p
p
p
and natural
n
n
n
such that
p
≥
n
≥
3
p \geq n \geq 3
p
≥
n
≥
3
. Set
A
A
A
is made of sequences of lenght
n
n
n
with elements from the set
{
0
,
1
,
2
,
.
.
.
,
p
−
1
}
\{0,1,2,...,p-1\}
{
0
,
1
,
2
,
...
,
p
−
1
}
and have the following property: For arbitrary two sequence
(
x
1
,
.
.
.
,
x
n
)
(x_1,...,x_n)
(
x
1
,
...
,
x
n
)
and
(
y
1
,
.
.
.
,
y
n
)
(y_1,...,y_n)
(
y
1
,
...
,
y
n
)
from the set
A
A
A
there exist three different numbers
k
,
l
,
m
k,l,m
k
,
l
,
m
such that:
x
k
≠
y
k
x_k \not = y_k
x
k
=
y
k
,
x
l
≠
y
l
x_l \not = y_l
x
l
=
y
l
,
x
m
≠
y
m
x_m \not = y_m
x
m
=
y
m
. Find the largest possible cardinality of
A
A
A
.
1
2
Hide problems
system of inequalities and equations
Positive integers
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
satisfy:
a
2
+
b
2
=
c
2
a^2+b^2=c^2
a
2
+
b
2
=
c
2
,
x
2
+
y
2
=
z
2
x^2+y^2=z^2
x
2
+
y
2
=
z
2
and
∣
x
−
a
∣
≤
1
|x-a| \leq 1
∣
x
−
a
∣
≤
1
,
∣
y
−
b
∣
≤
1
|y-b| \leq 1
∣
y
−
b
∣
≤
1
. Prove that sets
{
a
,
b
}
\{a,b\}
{
a
,
b
}
and
{
x
,
y
}
\{x,y\}
{
x
,
y
}
are equal.
bounded and periodic sequence
Let
c
c
c
be fixed natural number. Sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by:
a
1
=
1
a_1=1
a
1
=
1
,
a
n
+
1
=
d
(
a
n
)
+
c
a_{n+1}=d(a_n)+c
a
n
+
1
=
d
(
a
n
)
+
c
for
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
. where
d
(
m
)
d(m)
d
(
m
)
is number of divisors of
m
m
m
. Prove that there exist
k
k
k
natural such that sequence
a
k
,
a
k
+
1
,
.
.
.
a_k,a_{k+1},...
a
k
,
a
k
+
1
,
...
is periodic.