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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2005 Kurschak Competition
2005 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Domino towers
We build a tower of
2
×
1
2\times 1
2
×
1
dominoes in the following way. First, we place
55
55
55
dominoes on the table such that they cover a
10
×
11
10\times 11
10
×
11
rectangle; this is the first story of the tower. We then build every new level with
55
55
55
domioes above the exact same
10
×
11
10\times 11
10
×
11
rectangle. The tower is called stable if for every non-lattice point of the
10
×
11
10\times 11
10
×
11
rectangle, we can find a domino that has an inner point above it. How many stories is the lowest stable tower?
2
1
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Tennis
A and B play tennis. The player to first win at least four points and at least two more than the other player wins. We know that A gets a point each time with probability
p
≤
1
2
p\le \frac12
p
≤
2
1
, independent of the game so far. Prove that the probability that A wins is at most
2
p
2
2p^2
2
p
2
.
1
1
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Choice of indexes in a sequence
Let
N
>
1
N>1
N
>
1
and let
a
1
,
a
2
,
…
,
a
N
a_1,a_2,\dots,a_N
a
1
,
a
2
,
…
,
a
N
be nonnegative reals with sum at most
500
500
500
. Prove that there exist integers
k
≥
1
k\ge 1
k
≥
1
and
1
=
n
0
<
n
1
<
⋯
<
n
k
=
N
1=n_0<n_1<\dots<n_k=N
1
=
n
0
<
n
1
<
⋯
<
n
k
=
N
such that
∑
i
=
1
k
n
i
a
n
i
−
1
<
2005.
\sum_{i=1}^k n_ia_{n_{i-1}}<2005.
i
=
1
∑
k
n
i
a
n
i
−
1
<
2005.