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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2015 Kurschak Competition
2015 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Binary sequences with n terms
Let
Q
=
{
0
,
1
}
n
Q=\{0,1\}^n
Q
=
{
0
,
1
}
n
, and let
A
A
A
be a subset of
Q
Q
Q
with
2
n
−
1
2^{n-1}
2
n
−
1
elements. Prove that there are at least
2
n
−
1
2^{n-1}
2
n
−
1
pairs
(
a
,
b
)
∈
A
×
(
Q
∖
A
)
(a,b)\in A\times (Q\setminus A)
(
a
,
b
)
∈
A
×
(
Q
∖
A
)
for which sequences
a
a
a
and
b
b
b
differ in only one term.
2
1
Hide problems
Lines intersecting circles
Consider a triangle
A
B
C
ABC
A
BC
and a point
D
D
D
on its side
A
B
‾
\overline{AB}
A
B
. Let
I
I
I
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
on the angle bisector of
A
C
B
ACB
A
CB
. The second intersections of lines
A
I
AI
A
I
and
C
I
CI
C
I
with circle
A
C
D
ACD
A
C
D
are
P
P
P
and
Q
Q
Q
, respectively. Similarly, the second intersection of lines
B
I
BI
B
I
and
C
I
CI
C
I
with circle
B
C
D
BCD
BC
D
are
R
R
R
and
S
S
S
, respectively. Show that if
P
≠
Q
P\neq Q
P
=
Q
and
R
≠
S
R\neq S
R
=
S
, then lines
A
B
AB
A
B
,
P
Q
PQ
PQ
and
R
S
RS
RS
pass through a point or are parallel.
1
1
Hide problems
Fencing, probability of winning
In fencing, you win a round if you are the first to reach
15
15
15
points. Suppose that when
A
A
A
plays against
B
B
B
, at any point during the round,
A
A
A
scores the next point with probability
p
p
p
and
B
B
B
scores the next point with probability
q
=
1
−
p
q=1-p
q
=
1
−
p
. (However, they never can both score a point at the same time.)Suppose that in this round,
A
A
A
already has
14
−
k
14-k
14
−
k
points, and
B
B
B
has
14
−
ℓ
14-\ell
14
−
ℓ
(where
0
≤
k
,
ℓ
≤
14
0\le k,\ell\le 14
0
≤
k
,
ℓ
≤
14
). By how much will the probability that
A
A
A
wins the round increase if
A
A
A
scores the next point?