MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1975 Czech and Slovak Olympiad III A
1975 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
Real linear subspace closed under multiplication
Let
M
⊆
R
2
\mathbf M\subseteq\mathbb R^2
M
⊆
R
2
be a set with the following properties: 1) there is a pair
(
a
,
b
)
∈
M
(a,b)\in\mathbf M
(
a
,
b
)
∈
M
such that
a
b
(
a
−
b
)
≠
0
,
ab(a-b)\neq0,
ab
(
a
−
b
)
=
0
,
2) if
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
M
\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
∈
M
and
c
∈
R
c\in\mathbb R
c
∈
R
then also
(
c
x
1
,
c
y
1
)
,
(
x
1
+
x
2
,
y
1
+
y
2
)
,
(
x
1
x
2
,
y
1
y
2
)
∈
M
.
\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.
(
c
x
1
,
c
y
1
)
,
(
x
1
+
x
2
,
y
1
+
y
2
)
,
(
x
1
x
2
,
y
1
y
2
)
∈
M
.
Show that in fact
M
=
R
2
.
\mathbf M=\mathbb R^2.
M
=
R
2
.
5
1
Hide problems
Locus of vertices of isosceles triangles
Let a square
P
=
P
1
P
2
P
3
P
4
\mathbf P=P_1P_2P_3P_4
P
=
P
1
P
2
P
3
P
4
be given in the plane. Determine the locus of all vertices
A
A
A
of isosceles triangles
A
B
C
,
A
B
=
B
C
ABC,AB=BC
A
BC
,
A
B
=
BC
such that the vertices
B
,
C
B,C
B
,
C
are points of the square
P
.
\mathbf P.
P
.
4
1
Hide problems
Parametrization of a ray
Determine all real values of parameter
p
p
p
such that the equation
∣
x
−
2
∣
+
∣
y
−
3
∣
+
y
=
p
|x-2|+|y-3|+y=p
∣
x
−
2∣
+
∣
y
−
3∣
+
y
=
p
is an equation of a ray in the plane
x
y
.
xy.
x
y
.
3
1
Hide problems
Cyclic system with 6 variables
Determine all real tuples
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
)
\left(x_1,x_2,x_3,x_4,x_5,x_6\right)
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
)
such that \begin{align*} x_1(x_6 + x_2) &= x_3 + x_5, \\ x_2(x_1 + x_3) &= x_4 + x_6, \\ x_3(x_2 + x_4) &= x_5 + x_1, \\ x_4(x_3 + x_5) &= x_6 + x_2, \\ x_5(x_4 + x_6) &= x_1 + x_3, \\ x_6(x_5 + x_1) &= x_2 + x_4. \end{align*}
2
1
Hide problems
System of equations with squared floor function
Show that the system of equations \begin{align*} \lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\ 3x+y &=2, \end{align*} has infinitely many solutions and all these solutions satisfy bounds \begin{align*} 0<\ &x <4, \\ -9\le\ &y\le 1. \end{align*}
1
1
Hide problems
Acute triangle contained in right triangle
Let
T
\mathbf T
T
be a triangle with
[
T
]
=
1.
[\mathbf T]=1.
[
T
]
=
1.
Show that there is a right triangle
R
\mathbf R
R
such that
[
R
]
≤
3
[\mathbf R]\le\sqrt3
[
R
]
≤
3
and
T
⊆
R
.
\mathbf T\subseteq\mathbf R.
T
⊆
R
.
(
[
−
]
[-]
[
−
]
denotes area of a triangle.)