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Contests
International Contests
Czech-Polish-Slovak Match
2007 Czech-Polish-Slovak Match
2007 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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AB+CD≥BC+AD in convex ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. A circle passing through the points
A
A
A
and
D
D
D
and a circle passing through the points
B
B
B
and
C
C
C
are externally tangent at a point
P
P
P
inside the quadrilateral. Suppose that
∠
P
A
B
+
∠
P
D
C
≤
9
0
∘
\angle PAB+\angle PDC \leq 90^{\circ}
∠
P
A
B
+
∠
P
D
C
≤
9
0
∘
and
∠
P
B
A
+
∠
P
C
D
≤
9
0
∘
.
\angle PBA+\angle PCD \leq 90^{\circ}.
∠
PB
A
+
∠
PC
D
≤
9
0
∘
.
Prove that
A
B
+
C
D
≥
B
C
+
A
D
.
AB+CD\geq BC+AD.
A
B
+
C
D
≥
BC
+
A
D
.
3
1
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Cyclic quadrilateral ABCD with CD^2=AD.ED; E=DA∩CB
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
inscribed in a circle
k
k
k
has the property that the rays
D
A
DA
D
A
and
C
B
CB
CB
meet at a point
E
E
E
for which CD^2=AD\cdot ED. The perpendicular to
E
D
ED
E
D
at
A
A
A
intersects
k
k
k
again at point
F
.
F.
F
.
Prove that the segments
A
D
AD
A
D
and
C
F
CF
CF
are congruent if and only if the circumcenter of
△
A
B
E
\triangle ABE
△
A
BE
lies on
E
D
.
ED.
E
D
.
5
1
Hide problems
{1,2,...,n} partitioned into triplets such that a+b=c
For which
n
∈
{
3900
,
3901
,
⋯
,
3909
}
n\in\{3900, 3901,\cdots, 3909\}
n
∈
{
3900
,
3901
,
⋯
,
3909
}
can the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, . . . , n\}
{
1
,
2
,
...
,
n
}
be partitioned into (disjoint) triples in such a way that in each triple one of the numbers equals the sum of the other two?
4
1
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Real p≥1; ∃a,b,c,d∈S such that ab=cd
For any real number
p
≥
1
p\geq1
p
≥
1
consider the set of all real numbers
x
x
x
with
p
<
x
<
(
2
+
p
+
1
4
)
2
.
p<x<\left(2+\sqrt{p+\frac{1}{4}}\right)^2.
p
<
x
<
(
2
+
p
+
4
1
)
2
.
Prove that from any such set one can select four mutually distinct natural numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
with
a
b
=
c
d
.
ab=cd.
ab
=
c
d
.
2
1
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Fibonacci sequence=>∀m; ∃ k: m|(a_k^4-a_k-2)
The Fibonacci sequence is defined by
a
1
=
a
2
=
1
a_1=a_2=1
a
1
=
a
2
=
1
and
a
k
+
2
=
a
k
+
1
+
a
k
a_{k+2}=a_{k+1}+a_k
a
k
+
2
=
a
k
+
1
+
a
k
for
k
∈
N
.
k\in\mathbb N.
k
∈
N
.
Prove that for any natural number
m
,
m,
m
,
there exists an index
k
k
k
such that
a
k
4
−
a
k
−
2
a_k^4-a_k-2
a
k
4
−
a
k
−
2
is divisible by
m
.
m.
m
.
1
1
Hide problems
Polynomials P[R] such that P(x^2)=P(x)P(x+2)
Find all polynomials
P
P
P
with real coefficients satisfying
P
(
x
2
)
=
P
(
x
)
⋅
P
(
x
+
2
)
P(x^2)=P(x)\cdot P(x+2)
P
(
x
2
)
=
P
(
x
)
⋅
P
(
x
+
2
)
for all real numbers
x
.
x.
x
.