MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1968 Czech and Slovak Olympiad III A
1968 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
3
1
Hide problems
Locus given by both point and line reflection
Two segment
A
B
,
C
D
AB,CD
A
B
,
C
D
of the same length are given in plane such that lines
A
B
,
C
D
AB,CD
A
B
,
C
D
are not parallel. Consider a point
S
S
S
with the following property: the image of segment
A
B
AB
A
B
under point reflection with respect to
S
S
S
is identical to the mirror-image of segment
C
D
CD
C
D
with respect to some axis. Find the locus of all such points
S
.
S.
S
.
4
1
Hide problems
Feuerbach sphere
Four different points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are given in space such that
A
C
⊥
B
D
,
A
D
⊥
B
C
.
AC\perp BD,AD\perp BC.
A
C
⊥
B
D
,
A
D
⊥
BC
.
Show there is a sphere containing midpoits of all 7 segments
A
B
,
A
C
,
A
D
,
B
C
,
B
D
,
C
D
.
AB,AC,AD,BC,BD,CD.
A
B
,
A
C
,
A
D
,
BC
,
B
D
,
C
D
.
2
1
Hide problems
Powers of conjugates in Z[\sqrt3]
Show that for any integer
n
n
n
the number
a
n
=
(
2
+
3
)
n
−
(
2
−
3
)
n
2
3
a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}
a
n
=
2
3
(
2
+
3
)
n
−
(
2
−
3
)
n
is also integer. Determine all integers
n
n
n
such that
a
n
a_n
a
n
is divisible by 3.
1
1
Hide problems
System of equations and inequality
Let
a
1
,
…
,
a
n
(
n
>
2
)
a_1,\ldots,a_n\ (n>2)
a
1
,
…
,
a
n
(
n
>
2
)
be real numbers with at most one zero. Solve the system \begin{align*} x_1x_2 &= a_1, \\ x_2x_3 &= a_2, \\ &\ \vdots \\ x_{n-1}x_n &= a_{n-1}, \\ x_nx_1 &\ge a_n. \end{align*}