MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1954 Poland - Second Round
1954 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
sin (x_1 + x_2 + ...+ x_n) < sin x_1 + sin x_2 + ,,,+ sin x_n
Prove that if
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
are angles between
0
∘
0^\circ
0
∘
and
18
0
∘
180^\circ
18
0
∘
, and
n
n
n
is any natural number greater than
1
1
1
, then
sin
(
x
1
+
x
2
+
…
+
x
n
)
<
sin
x
1
+
sin
x
2
+
…
+
sin
x
n
.
\sin (x_1 + x_2 + \ldots + x_n) < \sin x_1 + \sin x_2 + \ldots + \sin x_n.
sin
(
x
1
+
x
2
+
…
+
x
n
)
<
sin
x
1
+
sin
x
2
+
…
+
sin
x
n
.
5
1
Hide problems
orthogonal projection of quad is parallelogram
Given points
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
that do not lie in the same plane. Draw a plane through the point
A
A
A
such that the orthogonal projection of the quadrilateral
A
B
C
D
ABCD
A
BC
D
on this plane is a parallelogram.
4
1
Hide problems
\sqrt{x - a} + \sqrt{x - b} = \sqrt{x - c }
Give the conditions under which the equation
x
−
a
+
x
−
b
=
x
−
c
\sqrt{x - a} + \sqrt{x - b} = \sqrt{x - c }
x
−
a
+
x
−
b
=
x
−
c
has roots, assuming that the numbers
a
a
a
,
b
b
b
,
c
c
c
are pairs of differences
3
1
Hide problems
triangle construction
Given: point
A
A
A
, line
p
p
p
, and circle
k
k
k
. Construct a triangle
A
B
C
ABC
A
BC
with angles
A
=
6
0
∘
A = 60^\circ
A
=
6
0
∘
,
B
=
9
0
∘
B = 90^\circ
B
=
9
0
∘
, whose vertex
B
B
B
lies on line
p
p
p
, and vertex
C
C
C
- on circle
k
k
k
.
2
1
Hide problems
among ten consecutive naturals
Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers
2
2
2
,
3
3
3
,
5
5
5
,
7
7
7
.
1
1
Hide problems
circle passing through touchpoints of n circles
The cross-section of a ball bearing consists of two concentric circles
C
C
C
and
C
1
C_1
C
1
, between which there are
n
n
n
small circles
k
1
,
k
2
,
…
,
k
n
k_1, k_2, \ldots, k_n
k
1
,
k
2
,
…
,
k
n
, each of which is tangent to the two adjacent circles and to both circles
C
C
C
and
C
1
C_1
C
1
. Given the radius
r
r
r
of the inner circle
C
C
C
and a natural number
n
n
n
, calculate the radius
x
x
x
of circle
C
2
C_2
C
2
passing through the points of tangency of circles
k
1
,
k
2
,
…
,
k
n
k_1, k_2, \ldots, k_n
k
1
,
k
2
,
…
,
k
n
and the sum
s
s
s
of the lengths of the arcs of circles
k
1
,
k
2
,
…
,
k
n
k_1, k_2, \ldots, k_n
k
1
,
k
2
,
…
,
k
n
that lie outside circle
C
2
C_2
C
2
.