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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2000 Czech and Slovak Match
2000 Czech and Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
1
1
Hide problems
a,b,c are sidelengths if 5abc>a^3+b^3+c^3
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers which satisfy
5
a
b
c
>
a
3
+
b
3
+
c
3
5abc>a^3+b^3+c^3
5
ab
c
>
a
3
+
b
3
+
c
3
. Prove that
a
,
b
,
c
a,b,c
a
,
b
,
c
can form a triangle.
6
1
Hide problems
2 odd integers of same color with difference x,y,x+y,x-y
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let
x
x
x
and
y
y
y
be odd integers such that
∣
x
∣
≠
∣
y
∣
|x| \ne |y|
∣
x
∣
=
∣
y
∣
. Show that there are two integers of the same color whose difference has one of the following values:
x
,
y
,
x
+
y
,
x
−
y
x,y,x+y,x-y
x
,
y
,
x
+
y
,
x
−
y
.
5
1
Hide problems
isosceles triangle wanted, circles in an isosceles trapezoid related
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid with bases
A
B
AB
A
B
and
C
D
CD
C
D
. The incircle of the triangle
B
C
D
BCD
BC
D
touches
C
D
CD
C
D
at
E
E
E
. Point
F
F
F
is chosen on the bisector of the angle
D
A
C
DAC
D
A
C
such that the lines
E
F
EF
EF
and
C
D
CD
C
D
are perpendicular. The circumcircle of the triangle
A
C
F
ACF
A
CF
intersects the line
C
D
CD
C
D
again at
G
G
G
. Prove that the triangle
A
F
G
AFG
A
FG
is isosceles.
4
1
Hide problems
P(x^4)P(x^3)P(x^2)P(x)+1 has no integer roots
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients. Prove that the polynomial
Q
(
x
)
=
P
(
x
4
)
P
(
x
3
)
P
(
x
2
)
P
(
x
)
+
1
Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1
Q
(
x
)
=
P
(
x
4
)
P
(
x
3
)
P
(
x
2
)
P
(
x
)
+
1
has no integer roots.
3
1
Hide problems
n power of 2 iff exists m so that 2^n-1 divides m^2 +9
Let
n
n
n
be a positive integer. Prove that
n
n
n
is a power of two if and only if there exists an integer
m
m
m
such that
2
n
−
1
2^n-1
2
n
−
1
is a divisor of
m
2
+
9
m^2 +9
m
2
+
9
.
2
1
Hide problems
incircle, circumcircles and 3 orthogonal circles
Let
A
B
C
{ABC}
A
BC
be a triangle,
k
{k}
k
its incircle and
k
a
,
k
b
,
k
c
{k_a,k_b,k_c}
k
a
,
k
b
,
k
c
three circles orthogonal to
k
{k}
k
passing through
B
{B}
B
and
C
,
A
{C, A}
C
,
A
and
C
{C}
C
, and
A
{A}
A
and
B
{B}
B
respectively. The circles
k
a
,
k
b
{k_a,k_b}
k
a
,
k
b
meet again in
C
′
{C'}
C
′
; in the same way we obtain the points
B
′
{B'}
B
′
and
A
′
{A'}
A
′
. Prove that the radius of the circumcircle of
A
′
B
′
C
′
{A'B'C'}
A
′
B
′
C
′
is half the radius of
k
{k}
k
.