MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2015 Poland - Second Round
2015 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
1
2
Hide problems
6 double lengths of segments given, parallelogram wanted
Points
E
,
F
,
G
E, F, G
E
,
F
,
G
lie, and on the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively of a triangle
A
B
C
ABC
A
BC
, with
2
A
G
=
G
B
,
2
B
E
=
E
C
2AG=GB, 2BE=EC
2
A
G
=
GB
,
2
BE
=
EC
and
2
C
F
=
F
A
2CF=FA
2
CF
=
F
A
. Points
P
P
P
and
Q
Q
Q
lie on segments
E
G
EG
EG
and
F
G
FG
FG
, respectively such that
2
E
P
=
P
G
2EP = PG
2
EP
=
PG
and
2
G
Q
=
Q
F
2GQ=QF
2
GQ
=
QF
. Prove that the quadrilateral
A
G
P
Q
AGPQ
A
GPQ
is a parallelogram.
x_3 + x_4 \in Q, x_3 x_4 \notin Q => x_1 + x_2 = x_3 + x_4
Real numbers
x
1
,
x
2
,
x
3
,
x
4
x_1, x_2, x_3, x_4
x
1
,
x
2
,
x
3
,
x
4
are roots of the fourth degree polynomial
W
(
x
)
W (x)
W
(
x
)
with integer coefficients. Prove that if
x
3
+
x
4
x_3 + x_4
x
3
+
x
4
is a rational number and
x
3
x
4
x_3x_4
x
3
x
4
is a irrational number, then
x
1
+
x
2
=
x
3
+
x
4
x_1 + x_2 = x_3 + x_4
x
1
+
x
2
=
x
3
+
x
4
.
2
2
Hide problems
Inequality
Let
A
A
A
be an integer and
A
>
1
A>1
A
>
1
. Let
a
1
=
A
A
a_{1}=A^{A}
a
1
=
A
A
,
a
n
+
1
=
A
a
n
a_{n+1}=A^{a_{n}}
a
n
+
1
=
A
a
n
and
b
1
=
A
A
+
1
b_{1}=A^{A+1}
b
1
=
A
A
+
1
,
b
n
+
1
=
2
b
n
b_{n+1}=2^{b_{n}}
b
n
+
1
=
2
b
n
,
n
=
1
,
2
,
3
,
.
.
.
n=1, 2, 3, ...
n
=
1
,
2
,
3
,
...
. Prove that
a
n
<
b
n
a_{n}<b_{n}
a
n
<
b
n
for each
n
n
n
.
no of sequences with terms in {0,1,2,3} so that n=a_0+2a_1+2^2a_2+...+2^na_n
Let
n
n
n
be a positive integer. Determine the number of sequences
a
0
,
a
1
,
…
,
a
n
a_0, a_1, \ldots, a_n
a
0
,
a
1
,
…
,
a
n
with terms in the set
{
0
,
1
,
2
,
3
}
\{0,1,2,3\}
{
0
,
1
,
2
,
3
}
such that
n
=
a
0
+
2
a
1
+
2
2
a
2
+
…
+
2
n
a
n
.
n=a_0+2a_1+2^2a_2+\ldots+2^na_n.
n
=
a
0
+
2
a
1
+
2
2
a
2
+
…
+
2
n
a
n
.
3
2
Hide problems
Number theory
Let
a
n
=
∣
n
(
n
+
1
)
−
19
∣
a_{n}=|n(n+1)-19|
a
n
=
∣
n
(
n
+
1
)
−
19∣
for
n
=
0
,
1
,
2
,
.
.
.
n=0, 1, 2, ...
n
=
0
,
1
,
2
,
...
and
n
≠
4
n \neq 4
n
=
4
. Prove that if for every
k
<
n
k<n
k
<
n
we have
gcd
(
a
n
,
a
k
)
=
1
\gcd(a_{n}, a_{k})=1
g
cd
(
a
n
,
a
k
)
=
1
, then
a
n
a_{n}
a
n
is a prime number.
Geometry problem
Let
A
B
C
ABC
A
BC
be a triangle. Let
K
K
K
be a midpoint of
B
C
BC
BC
and
M
M
M
be a point on the segment
A
B
AB
A
B
.
L
=
K
M
∩
A
C
L=KM \cap AC
L
=
K
M
∩
A
C
and
C
C
C
lies on the segment
A
C
AC
A
C
between
A
A
A
and
L
L
L
. Let
N
N
N
be a midpoint of
M
L
ML
M
L
.
A
N
AN
A
N
cuts circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
in
S
S
S
and
S
≠
N
S \neq N
S
=
N
. Prove that circumcircle of
Δ
K
S
N
\Delta KSN
Δ
K
SN
is tangent to
B
C
BC
BC
.