MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2012 Kurschak Competition
2012 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
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Angles and excenters
Let
J
A
J_A
J
A
and
J
B
J_B
J
B
be the
A
A
A
-excenter and
B
B
B
-excenter of
△
A
B
C
\triangle ABC
△
A
BC
. Consider a chord
P
Q
‾
\overline{PQ}
PQ
of circle
A
B
C
ABC
A
BC
which is parallel to
A
B
AB
A
B
and intersects segments
A
C
‾
\overline{AC}
A
C
and
B
C
‾
\overline{BC}
BC
. If lines
A
B
AB
A
B
and
C
P
CP
CP
intersect at
R
R
R
, prove that
∠
J
A
Q
J
B
+
∠
J
A
R
J
B
=
18
0
∘
.
\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.
∠
J
A
Q
J
B
+
∠
J
A
R
J
B
=
18
0
∘
.
3
1
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Probability of n events not happening
Consider
n
n
n
events, each of which has probability
1
2
\frac12
2
1
. We also know that the probability of any two both happening is
1
4
\frac14
4
1
. Prove the following. (a) The probability that none of these events happen is at most
1
n
+
1
\frac1{n+1}
n
+
1
1
. (b) We can reach equality in (a) for infinitely many
n
n
n
.
2
1
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Consecutive 2-Niven numbers
Denote by
E
(
n
)
E(n)
E
(
n
)
the number of
1
1
1
's in the binary representation of a positive integer
n
n
n
. Call
n
n
n
interesting if
E
(
n
)
E(n)
E
(
n
)
divides
n
n
n
. Prove that (a) there cannot be five consecutive interesting numbers, and (b) there are infinitely many positive integers
n
n
n
such that
n
n
n
,
n
+
1
n+1
n
+
1
and
n
+
2
n+2
n
+
2
are each interesting.