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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1952 Kurschak Competition
1952 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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perimeter of A'B'C' less than k times the perimeter of ABC
A
B
C
ABC
A
BC
is a triangle. The point A' lies on the side opposite to
A
A
A
and
B
A
′
/
B
C
=
k
BA'/BC = k
B
A
′
/
BC
=
k
, where
1
/
2
<
k
<
1
1/2 < k < 1
1/2
<
k
<
1
. Similarly,
B
′
B'
B
′
lies on the side opposite to
B
B
B
with
C
B
′
/
C
A
=
k
CB'/CA = k
C
B
′
/
C
A
=
k
, and
C
′
C'
C
′
lies on the side opposite to
C
C
C
with
A
C
′
/
A
B
=
k
AC'/AB = k
A
C
′
/
A
B
=
k
. Show that the perimeter of
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is less than
k
k
k
times the perimeter of
A
B
C
ABC
A
BC
.
2
1
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2 from n + 2 numbers from set 1-3n with n < difference <2n
Show that if we choose any
n
+
2
n + 2
n
+
2
distinct numbers from the set
{
1
,
2
,
3
,
.
.
.
,
3
n
}
\{1, 2, 3, . . . , 3n\}
{
1
,
2
,
3
,
...
,
3
n
}
there will be two whose difference is greater than
n
n
n
and smaller than
2
n
2n
2
n
.
1
1
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circle touches three pairwise disjoint circles with collinear centers
A circle
C
C
C
touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.