MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1953 Poland - Second Round
1953 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
circle construction
Given a circle and two tangents to this circle. Draw a third tangent to the circle in such a way that its segment contained by the given tangents has the given length
d
d
d
.
5
1
Hide problems
volume of tetrahedron
Calculate the volume
V
V
V
of tetrahedron
A
B
C
D
ABCD
A
BC
D
given the length
d
d
d
of edge
A
B
AB
A
B
and the area
S
S
S
of the projection of the tetrahedron on the plane perpendicular to the line
A
B
AB
A
B
.
4
1
Hide problems
x_k x_{k+1} = k, nxn system
Solve the system of equations
<
b
r
/
>
<
b
r
/
>
x
1
x
2
=
1
<
b
r
/
>
x
2
x
3
=
2
<
b
r
/
>
x
3
x
4
=
3
<
b
r
/
>
…
<
b
r
/
>
x
n
x
1
=
n
<
b
r
/
>
\qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}
<
b
r
/
>
<
b
r
/
>
x
1
x
2
=
1
<
b
r
/
>
x
2
x
3
=
2
<
b
r
/
>
x
3
x
4
=
3
<
b
r
/
>
…
<
b
r
/
>
x
n
x
1
=
n
<
b
r
/
>
3
1
Hide problems
triangular piece of sheet metal weighs
A triangular piece of sheet metal weighs
900
900
900
g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than
400
400
400
g.
2
1
Hide problems
10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64
The board was placed
<
b
r
/
>
1
=
1
<
b
r
/
>
2
+
3
+
4
=
1
+
8
<
b
r
/
>
5
+
6
+
7
+
8
+
9
=
8
+
27
<
b
r
/
>
10
+
11
+
12
+
13
+
14
+
15
+
16
=
27
+
64
<
b
r
/
>
…
<
b
r
/
>
\begin{array}{rcl}<br /> 1 & = & 1 \\<br /> 2 + 3 + 4 & = & 1 + 8 \\<br /> 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\<br /> 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\<br /> & \ldots &<br /> \end{array}
<
b
r
/
>
1
<
b
r
/
>
2
+
3
+
4
<
b
r
/
>
5
+
6
+
7
+
8
+
9
<
b
r
/
>
10
+
11
+
12
+
13
+
14
+
15
+
16
<
b
r
/
>
=
=
=
=
…
1
1
+
8
8
+
27
27
+
64
<
b
r
/
>
Write such a formula for the
n
n
n
-th row of the array that, with the substitutions
n
=
1
,
2
,
3
,
4
n = 1, 2, 3, 4
n
=
1
,
2
,
3
,
4
, would give the above four lines of the array and would be true for every natural
n
n
n
.
1
1
Hide problems
(x - a) (x - c) + 2 (x - b) (x - d) = 0
Prove that the equation
(
x
−
a
)
(
x
−
c
)
+
2
(
x
−
b
)
(
x
−
d
)
=
0
,
(x - a) (x - c) + 2 (x - b) (x - d) = 0,
(
x
−
a
)
(
x
−
c
)
+
2
(
x
−
b
)
(
x
−
d
)
=
0
,
in which
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
, has two real roots.