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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1963 Kurschak Competition
1963 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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sum of the medians is greater than four times the circumradius
A triangle has no angle greater than
9
0
o
90^o
9
0
o
. Show that the sum of the medians is greater than four times the circumradius.
2
1
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(1 +1/sen A ) (1 +1/cos A) > 5 for acute <A
A
A
A
is an acute angle. Show that
(
1
+
1
s
e
n
A
)
(
1
+
1
c
o
s
A
)
>
5
\left(1 +\frac{1}{sen A}\right)\left(1 +\frac{1}{cos A}\right)> 5
(
1
+
se
n
A
1
)
(
1
+
cos
A
1
)
>
5
1
1
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mn students all have different heights
m
n
mn
mn
students all have different heights. They are arranged in
m
>
1
m > 1
m
>
1
rows of
n
>
1
n > 1
n
>
1
. In each row select the shortest student and let
A
A
A
be the height of the tallest such. In each column select the tallest student and let
B
B
B
be the height of the shortest such. Which of the following are possible:
A
<
B
A < B
A
<
B
,
A
=
B
A = B
A
=
B
,
A
>
B
A > B
A
>
B
? If a relation is possible, can it always be realized by a suitable arrangement of the students?