MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2012 Poland - Second Round
2012 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
3
2
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Divisor of a product
Let
m
,
n
∈
Z
+
m,n\in\mathbb{Z_{+}}
m
,
n
∈
Z
+
be such numbers that set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
contains exactly
m
m
m
different prime numbers. Prove that if we choose any
m
+
1
m+1
m
+
1
different numbers from
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
then we can find number from
m
+
1
m+1
m
+
1
choosen numbers, which divide product of other
m
m
m
numbers.
Sum of digits
Denote by
S
(
k
)
S(k)
S
(
k
)
the sum of the digits in the decimal representation of
k
k
k
. Prove that there are infinitely many
n
∈
Z
+
n\in \mathbb{Z_{+}}
n
∈
Z
+
such that:
S
(
2
n
+
n
)
<
S
(
2
n
)
{S(2^{n}+n})<S(2^{n})
S
(
2
n
+
n
)
<
S
(
2
n
)
.
2
2
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Tetrahedron
Prove that for tetrahedron
A
B
C
D
ABCD
A
BC
D
; vertex
D
D
D
, center of insphere and centroid of
A
B
C
D
ABCD
A
BC
D
are collinear iff areas of triangles
A
B
D
,
B
C
D
,
C
A
D
ABD,BCD,CAD
A
B
D
,
BC
D
,
C
A
D
are equal.
Common point
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
∘
\angle A=60^{\circ}
∠
A
=
6
0
∘
and
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
,
I
I
I
-incenter,
O
O
O
-circumcenter. Prove that perpendicular bisector of
A
I
AI
A
I
, line
O
I
OI
O
I
and line
B
C
BC
BC
have a common point.
1
2
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System of equations
a
,
b
,
c
,
d
∈
R
a,b,c,d\in\mathbb{R}
a
,
b
,
c
,
d
∈
R
, solve the system of equations:
{
a
3
+
b
=
c
b
3
+
c
=
d
c
3
+
d
=
a
d
3
+
a
=
b
\begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases}
⎩
⎨
⎧
a
3
+
b
=
c
b
3
+
c
=
d
c
3
+
d
=
a
d
3
+
a
=
b
Functional equation
f
,
g
:
R
→
R
f,g:\mathbb{R}\rightarrow\mathbb{R}
f
,
g
:
R
→
R
find all
f
,
g
f,g
f
,
g
satisfying
∀
x
,
y
∈
R
\forall x,y\in \mathbb{R}
∀
x
,
y
∈
R
:
g
(
f
(
x
)
−
y
)
=
f
(
g
(
y
)
)
+
x
.
g(f(x)-y)=f(g(y))+x.
g
(
f
(
x
)
−
y
)
=
f
(
g
(
y
))
+
x
.