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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1943 Eotvos Mathematical Competition
1943 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
3
1
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min max (x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2
Let
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
be real numbers and
(
x
,
y
,
z
,
t
)
(x,y, z,t)
(
x
,
y
,
z
,
t
)
be any permutation of
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
. What are the maximum and minimum values of the expression
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
t
)
2
+
(
t
−
x
)
2
?
(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
t
)
2
+
(
t
−
x
)
2
?
2
1
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D >= 2d, max and min distance of a point from perimeter of acute triangle
Let
P
P
P
be any point inside an acute triangle. Let
D
D
D
and
d
d
d
be respectively the maximum and minimum distances from
P
P
P
to any point on the perimeter of the triangle. (a) Prove that
D
≥
2
d
D \ge 2d
D
≥
2
d
. (b) Determine when equality holds
1
1
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no of those who know an odd no of others in the group is even
Prove that in any group of people, the number of those who know an odd number of the others in the group is even. Assume that knowing is a symmetric relation.