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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2011 Czech-Polish-Slovak Match
2011 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(3)
2
2
Hide problems
Changing numbers on a blackboard to zeroes
Written on a blackboard are
n
n
n
nonnegative integers whose greatest common divisor is
1
1
1
. A move consists of erasing two numbers
x
x
x
and
y
y
y
, where
x
≥
y
x\ge y
x
≥
y
, on the blackboard and replacing them with the numbers
x
−
y
x-y
x
−
y
and
2
y
2y
2
y
. Determine for which original
n
n
n
-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where
n
−
1
n-1
n
−
1
of the numbers of the blackboard are zeroes.
Concurrent lines through midpoints in a quadrilateral
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, let
M
M
M
and
N
N
N
denote the midpoints of sides
A
D
AD
A
D
and
B
C
BC
BC
, respectively. On sides
A
B
AB
A
B
and
C
D
CD
C
D
are points
K
K
K
and
L
L
L
, respectively, such that
∠
M
K
A
=
∠
N
L
C
\angle MKA=\angle NLC
∠
M
K
A
=
∠
N
L
C
. Prove that if lines
B
D
BD
B
D
,
K
M
KM
K
M
, and
L
N
LN
L
N
are concurrent, then
∠
K
M
N
=
∠
B
D
C
and
∠
L
N
M
=
∠
A
B
D
.
\angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.
∠
K
MN
=
∠
B
D
C
and
∠
L
NM
=
∠
A
B
D
.
3
2
Hide problems
Three of four points on a circle determine a 108° angle
Points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
lie on a circle (in that order) where
A
B
AB
A
B
and
C
D
CD
C
D
are not parallel. The length of arc
A
B
AB
A
B
(which contains the points
D
D
D
and
C
C
C
) is twice the length of arc
C
D
CD
C
D
(which does not contain the points
A
A
A
and
B
B
B
). Let
E
E
E
be a point where
A
C
=
A
E
AC=AE
A
C
=
A
E
and
B
D
=
B
E
BD=BE
B
D
=
BE
. Prove that if the perpendicular line from point
E
E
E
to the line
A
B
AB
A
B
passes through the center of the arc
C
D
CD
C
D
(which does not contain the points
A
A
A
and
B
B
B
), then
∠
A
C
B
=
10
8
∘
\angle ACB = 108^\circ
∠
A
CB
=
10
8
∘
.
For a in Z, there are ∞ primes p s.t. p | n^2+3 & p | m^3-a
Let
a
a
a
be any integer. Prove that there are infinitely many primes
p
p
p
such that
p
∣
n
2
+
3
and
p
∣
m
3
−
a
p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a
p
∣
n
2
+
3
and
p
∣
m
3
−
a
for some integers
n
n
n
and
m
m
m
.
1
2
Hide problems
Ineq: a^2 < bc implies b^3 + ac^2 > ab(a+c)
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers satisfying
a
2
<
b
c
a^2<bc
a
2
<
b
c
. Prove that
b
3
+
a
c
2
>
a
b
(
a
+
c
)
b^3+ac^2>ab(a+c)
b
3
+
a
c
2
>
ab
(
a
+
c
)
.
Factoring a composition of two polynomials
A polynomial
P
(
x
)
P(x)
P
(
x
)
with integer coefficients satisfies the following: if
F
(
x
)
F(x)
F
(
x
)
,
G
(
x
)
G(x)
G
(
x
)
, and
Q
(
x
)
Q(x)
Q
(
x
)
are polynomials with integer coefficients satisfying
P
(
Q
(
x
)
)
=
F
(
x
)
⋅
G
(
x
)
P\Big(Q(x)\Big)=F(x)\cdot G(x)
P
(
Q
(
x
)
)
=
F
(
x
)
⋅
G
(
x
)
, then
F
(
x
)
F(x)
F
(
x
)
or
G
(
x
)
G(x)
G
(
x
)
is a constant polynomial. Prove that
P
(
x
)
P(x)
P
(
x
)
is a constant polynomial.