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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1961 Kurschak Competition
1961 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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concurrent wanted, 2 circles each ecternal to other one related
Two circles centers
O
O
O
and
O
′
O'
O
′
are disjoint.
P
P
′
PP'
P
P
′
is an outer tangent (with
P
P
P
on the circle center O, and P' on the circle center
O
′
O'
O
′
). Similarly,
Q
Q
′
QQ'
Q
Q
′
is an inner tangent (with
Q
Q
Q
on the circle center
O
O
O
, and
Q
′
Q'
Q
′
on the circle center
O
′
O'
O
′
). Show that the lines
P
Q
PQ
PQ
and
P
′
Q
′
P'Q'
P
′
Q
′
meet on the line
O
O
′
OO'
O
O
′
. https://cdn.artofproblemsolving.com/attachments/b/d/bad305631571323a61b097f149a1bb6855cdc5.png
2
1
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at least one of (1 - x)y, $(1 - y)z, (1 - z)x <= 1/4 if 0<x,y,z<1
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive reals less than
1
1
1
. Show that at least one of
(
1
−
x
)
y
(1 - x)y
(
1
−
x
)
y
,
(
1
−
y
)
z
(1 - y)z
(
1
−
y
)
z
and
(
1
−
z
)
x
(1 - z)x
(
1
−
z
)
x
does not exceed
1
4
\frac14
4
1
.
1
1
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ratio of max/min distance of any 4 points >=\sqrt2
Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least
2
\sqrt2
2
.