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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1967 Czech and Slovak Olympiad III A
1967 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
3
1
Hide problems
Products of cyclic permutations
Consider a table of cyclic permutations (
n
≥
2
n\ge2
n
≥
2
)
1
,
2
,
…
,
n
−
1
,
n
2
,
3
,
…
,
n
,
1
,
⋮
⋮
⋱
⋮
⋮
n
,
1
,
…
,
n
−
2
,
n
−
1.
\begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix}
1
,
2
,
⋮
n
,
2
,
3
,
⋮
1
,
…
,
…
,
⋱
…
,
n
−
1
,
n
,
⋮
n
−
2
,
n
1
,
⋮
n
−
1.
Then multiply each number of the first row by that number of the
k
k
k
-th row that is in the same column. Sum all these products and denote
s
k
s_k
s
k
the result (e.g.
s
2
=
1
⋅
2
+
2
⋅
3
+
⋯
+
(
n
−
1
)
⋅
n
+
n
⋅
1
s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1
s
2
=
1
⋅
2
+
2
⋅
3
+
⋯
+
(
n
−
1
)
⋅
n
+
n
⋅
1
). a) Find a recursive relation for
s
k
s_k
s
k
in terms of
s
k
−
1
s_{k-1}
s
k
−
1
and determine the explicit formula for
s
k
s_k
s
k
. b) Determine both an index
k
k
k
and the value of
s
k
s_k
s
k
such that the sum
s
k
s_k
s
k
is minimal.
4
1
Hide problems
Circumcircle and exterior line
Let
A
B
C
ABC
A
BC
be an acute triangle,
k
k
k
its circumcirle and
m
m
m
a line such that
m
∩
k
=
∅
,
m
∥
B
C
.
m\cap k=\emptyset, m\parallel BC.
m
∩
k
=
∅
,
m
∥
BC
.
Denote
D
D
D
the intersection of
m
m
m
and ray
A
B
.
AB.
A
B
.
a) Let
X
X
X
be an inner point of the arc
B
C
BC
BC
not containing
A
A
A
and denote
Y
Y
Y
the intersection of lines
m
,
C
X
.
m,CX.
m
,
CX
.
Show that
A
,
D
,
X
,
Y
A,D,X,Y
A
,
D
,
X
,
Y
are concyclic and name this circle
κ
\kappa
κ
. b) Determine relative position of
κ
\kappa
κ
and
m
m
m
in case when
C
,
D
,
X
C,D,X
C
,
D
,
X
are collinear.
2
1
Hide problems
Tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron such that
A
B
2
+
C
D
2
=
A
C
2
+
B
D
2
=
A
D
2
+
B
C
2
.
AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.
A
B
2
+
C
D
2
=
A
C
2
+
B
D
2
=
A
D
2
+
B
C
2
.
Show that at least one of its faces is an acute triangle.
1
1
Hide problems
Quartic equation
Find all triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of complex numbers such that the equation
x
4
−
a
x
3
−
b
x
+
c
=
0
x^4-ax^3-bx+c=0
x
4
−
a
x
3
−
b
x
+
c
=
0
has
a
,
b
,
c
a,b,c
a
,
b
,
c
as roots.