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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2005 Czech And Slovak Olympiad III A
2005 Czech And Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
1
1
Hide problems
for some k > 1: x_{k-1}y_{k-1} = 42, x_ky_k = 30, and x_{k+1}y_{k+1} = 16
Consider all arithmetical sequences of real numbers
(
x
i
)
∞
=
1
(x_i)^{\infty}=1
(
x
i
)
∞
=
1
and
(
y
i
)
∞
=
1
(y_i)^{\infty} =1
(
y
i
)
∞
=
1
with the common first term, such that for some
k
>
1
,
x
k
−
1
y
k
−
1
=
42
,
x
k
y
k
=
30
k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30
k
>
1
,
x
k
−
1
y
k
−
1
=
42
,
x
k
y
k
=
30
, and
x
k
+
1
y
k
+
1
=
16
x_{k+1}y_{k+1} = 16
x
k
+
1
y
k
+
1
=
16
. Find all such pairs of sequences with the maximum possible
k
k
k
.
5
1
Hide problems
x^2 + px+q = 0, x^2 +rx+s = 0 => pr = (q+1)(s+1) , p(q+1)s = r(s+1)q
Let
p
,
q
,
r
,
s
p,q, r, s
p
,
q
,
r
,
s
be real numbers with
q
≠
−
1
q \ne -1
q
=
−
1
and
s
≠
−
1
s \ne -1
s
=
−
1
. Prove that the quadratic equations
x
2
+
p
x
+
q
=
0
x^2 + px+q = 0
x
2
+
p
x
+
q
=
0
and
x
2
+
r
x
+
s
=
0
x^2 +rx+s = 0
x
2
+
r
x
+
s
=
0
have a common root, while their other roots are inverse of each other, if and only if
p
r
=
(
q
+
1
)
(
s
+
1
)
pr = (q+1)(s+1)
p
r
=
(
q
+
1
)
(
s
+
1
)
and
p
(
q
+
1
)
s
=
r
(
s
+
1
)
q
p(q+1)s = r(s+1)q
p
(
q
+
1
)
s
=
r
(
s
+
1
)
q
. (A double root is counted twice.)
6
1
Hide problems
numbers of each color from 4 form a monotone subsequence
Decide whether for every arrangement of the numbers
1
,
2
,
3
,
.
.
.
,
15
1,2,3, . . . ,15
1
,
2
,
3
,
...
,
15
in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.
3
1
Hide problems
tangent quadrilaterals ABED and AECD in trapezoid ABCD
In a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
/
/
C
D
,
E
AB // CD, E
A
B
//
C
D
,
E
is the midpoint of
B
C
BC
BC
. Prove that if the quadrilaterals
A
B
E
D
ABED
A
BE
D
and
A
E
C
D
AECD
A
EC
D
are tangent, then the sides
a
=
A
B
,
b
=
B
C
,
c
=
C
D
,
d
=
D
A
a = AB, b = BC, c =CD, d = DA
a
=
A
B
,
b
=
BC
,
c
=
C
D
,
d
=
D
A
of the trapezoid satisfy the equalities
a
+
c
=
b
3
+
d
a+c = \frac{b}{3} +d
a
+
c
=
3
b
+
d
and
1
a
+
1
c
=
3
b
\frac1a +\frac1c = \frac3b
a
1
+
c
1
=
b
3
.
2
1
Hide problems
2^{15} subsets X of {1,2,...,47}, min m in X, x in X, either x+m in X or x+m>47
Determine for which
m
m
m
there exist exactly
2
15
2^{15}
2
15
subsets
X
X
X
of
{
1
,
2
,
.
.
.
,
47
}
\{1,2,...,47\}
{
1
,
2
,
...
,
47
}
with the following property:
m
m
m
is the smallest element of
X
X
X
, and for every
x
∈
X
x \in X
x
∈
X
, either
x
+
m
∈
X
x+m \in X
x
+
m
∈
X
or
x
+
m
>
47
x+m > 47
x
+
m
>
47
.
4
1
Hide problems
locus of intersection of diagonals of rectangle circumscribed to a triagnle
An acute-angled triangle
A
K
L
AKL
A
K
L
is given on a plane. Consider all rectangles
A
B
C
D
ABCD
A
BC
D
circumscribed to triangle
A
K
L
AKL
A
K
L
such that point
K
K
K
lies on side
B
C
BC
BC
and point
L
L
L
lieson side
C
D
CD
C
D
. Find the locus of the intersection
S
S
S
of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
.