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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1970 Yugoslav Team Selection Test
1970 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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quadrilateral with sphere tg to edges
If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.
Problem 1
1
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NT inequality, number of digits
Positive integers
a
a
a
and
b
b
b
have
n
n
n
digits each in their decimal representation. Assume that
m
m
m
is a positive integer such that
n
2
<
m
<
n
\frac n2<m<n
2
n
<
m
<
n
and assume that each of the leftmost
m
m
m
digits of
a
a
a
is equal to the corresponding digit of
b
b
b
. Prove that
a
1
n
−
b
1
n
<
1
n
.
a^{\frac1n}-b^{\frac1n}<\frac1n.
a
n
1
−
b
n
1
<
n
1
.
Problem 2
1
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triangle, vertices on cube
Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.