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National and Regional Contests
Poland Contests
Poland - Second Round
1976 Poland - Second Round
1976 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(4)
6
1
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6 points on plane, each 3 are vertices of triangles with different sides
Six points are placed on the plane such that each three of them are the vertices of a triangle with sides of different lengths. Prove that the shortest side of one of these triangles is also the longest side of another of them.
5
1
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cos \pi x =1/3 then x is irrational
Prove that if
cos
π
x
=
1
3
\cos \pi x =\frac{1}{3}
cos
π
x
=
3
1
then
x
x
x
is an irrational number.
4
1
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concyclic points
Inside the circle
S
S
S
there is a circle
T
T
T
and circles
K
1
,
K
2
,
…
,
K
n
K_1, K_2, \ldots, K_n
K
1
,
K
2
,
…
,
K
n
tangent externally to
T
T
T
and internally to
S
S
S
, and the circle
K
1
K_1
K
1
is tangent to
K
2
K_2
K
2
,
K
2
K_2
K
2
tangent to
K
3
K_3
K
3
etc. Prove that the points of tangency of the circles
K
1
K_1
K
1
with
K
2
K_2
K
2
,
K
2
K_2
K
2
with
K
3
K_3
K
3
etc. lie on the circle.
3
1
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there is no isometry mapping this spherical bowl to a subset of the plane.
We consider a spherical bowl without any great circle. The distance between points
A
A
A
and
B
B
B
on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points
A
A
A
and
B
B
B
, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane.Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.