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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1973 Yugoslav Team Selection Test
1973 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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points covered ink, vector translation
Several points are denoted on a white piece of paper. The distance between each two of the points is greater than
24
24
24
. A drop of ink was sprinkled over the paper covering an area smaller than
π
\pi
π
. Prove that there exists a vector
v
→
\overrightarrow v
v
with
v
→
<
1
\overrightarrow v<1
v
<
1
, such that after translating all of the points by
v
v
v
none of them is covered in ink.
Problem 2
1
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given point, find diameter
A circle
k
k
k
is drawn using a given disc (e.g. a coin). A point
A
A
A
is chosen on
k
k
k
. Using just the given disc, determine the point
B
B
B
on
k
k
k
so that
A
B
AB
A
B
is a diameter of
k
k
k
. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)
Problem 1
1
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rectangle with integer sides
All sides of a rectangle are odd positive integers. Prove that there does not exist a point inside the rectangle whose distance to each of the vertices is an integer.