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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1959 Poland - Second Round
1959 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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collinearity with a sphere
From a point
M
M
M
on the surface of a sphere, three mutually perpendicular chords
M
A
MA
M
A
,
M
B
MB
MB
,
M
C
MC
MC
are drawn. Prove that the segment joining the point
M
M
M
with the center of the sphere intersects the plane of the triangle
A
B
C
ABC
A
BC
at the center of gravity of this triangle.
5
1
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every 3 of n segments have a common point.
In the plane,
n
≥
3
n \geq 3
n
≥
3
segments are placed in such a way that every
3
3
3
of them have a common point. Prove that there is a common point for all the segments.
4
1
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a_n =3(n^2 + n) + 7 , NT
Given a sequence of numbers
13
,
25
,
43
,
…
13, 25, 43, \ldots
13
,
25
,
43
,
…
whose
n
n
n
-th term is defined by the formula
a
n
=
3
(
n
2
+
n
)
+
7
a_n =3(n^2 + n) + 7
a
n
=
3
(
n
2
+
n
)
+
7
Prove that this sequence has the following properties:1) Of every five consecutive terms of the sequence, exactly one is divisible by
5
5
5
,2( No term of the sequence is the cube of an integer.
3
1
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tg \frac{a + b}{2} \leq \frac{tg a+ tg b}{2
Prove that if
0
≤
α
<
π
2
0 \leq \alpha < \frac{\pi}{2}
0
≤
α
<
2
π
and
0
≤
β
<
π
2
0 \leq \beta < \frac{\pi}{2}
0
≤
β
<
2
π
, then
t
g
α
+
β
2
≤
t
g
α
+
t
g
β
2
.
tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.
t
g
2
α
+
β
≤
2
t
gα
+
t
g
β
.
2
1
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triangle by medians of triangle
What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?
1
1
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ax^3 + bx^2 + cx + d = 0
What necessary and sufficient condition should the coefficients
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
satisfy so that the equation
a
x
3
+
b
x
2
+
c
x
+
d
=
0
ax^3 + bx^2 + cx + d = 0
a
x
3
+
b
x
2
+
c
x
+
d
=
0
has two opposite roots?