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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1986 Kurschak Competition
1986 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(2)
2
1
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Variation on Shapiro's
Let
n
>
2
n>2
n
>
2
be a positive integer. Find the largest value
h
h
h
and the smallest value
H
H
H
for which
h
<
a
1
a
1
+
a
2
+
a
2
a
2
+
a
3
+
⋯
+
a
n
a
n
+
a
1
<
H
h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H
h
<
a
1
+
a
2
a
1
+
a
2
+
a
3
a
2
+
⋯
+
a
n
+
a
1
a
n
<
H
holds for any positive reals
a
1
,
…
,
a
n
a_1,\dots,a_n
a
1
,
…
,
a
n
.
3
1
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Parity of the sum of randomly chosen integers
A and B plays the following game: they choose randomly
k
k
k
integers from
{
1
,
2
,
…
,
100
}
\{1,2,\dots,100\}
{
1
,
2
,
…
,
100
}
; if their sum is even, A wins, else B wins. For what values of
k
k
k
does A and B have the same chance of winning?