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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1990 Kurschak Competition
1990 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Fair coin-tossing
We would like to give a present to one of
100
100
100
children. We do this by throwing a biased coin
k
k
k
times, after predetermining who wins in each possible outcome of this lottery.Prove that we can choose the probability
p
p
p
of throwing heads, and the value of
k
k
k
such that, by distributing the
2
k
2^k
2
k
different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
2
1
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A concurrency with excenters
The incenter of
△
A
1
A
2
A
3
\triangle A_1A_2A_3
△
A
1
A
2
A
3
is
I
I
I
, and the center of the
A
i
A_i
A
i
-excircle is
J
i
J_i
J
i
(
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
). Let
B
i
B_i
B
i
be the intersection point of side
A
i
+
1
A
i
+
2
A_{i+1}A_{i+2}
A
i
+
1
A
i
+
2
and the bisector of
∠
A
i
+
1
I
A
i
+
2
\angle A_{i+1}IA_{i+2}
∠
A
i
+
1
I
A
i
+
2
(
A
i
+
3
:
=
A
i
A_{i+3}:=A_i
A
i
+
3
:=
A
i
∀
i
\forall i
∀
i
). Prove that the three lines
B
i
J
i
B_iJ_i
B
i
J
i
are concurrent.
1
1
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Divisor d of pn^2, such that n^2+d is a square
Let
p
>
2
p>2
p
>
2
be a prime number and
n
n
n
a positive integer. Prove that
p
n
2
pn^2
p
n
2
has at most one positive divisor
d
d
d
for which
n
2
+
d
n^2+d
n
2
+
d
is a square number.