MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2013 Kurschak Competition
2013 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Construction of a sequence, with a lower bound on |a_i-a_j|
Is it true that for integer
n
≥
2
n\ge 2
n
≥
2
, and given any non-negative reals
ℓ
i
j
\ell_{ij}
ℓ
ij
,
1
≤
i
<
j
≤
n
1\le i<j\le n
1
≤
i
<
j
≤
n
, we can find a sequence
0
≤
a
1
,
a
2
,
…
,
a
n
0\le a_1,a_2,\ldots,a_n
0
≤
a
1
,
a
2
,
…
,
a
n
such that for all
1
≤
i
<
j
≤
n
1\le i<j\le n
1
≤
i
<
j
≤
n
to have
∣
a
i
−
a
j
∣
≥
ℓ
i
j
|a_i-a_j|\ge \ell_{ij}
∣
a
i
−
a
j
∣
≥
ℓ
ij
, yet still
∑
i
=
1
n
a
i
≤
∑
1
≤
i
<
j
≤
n
ℓ
i
j
\sum_{i=1}^n a_i\le \sum_{1\le i<j\le n}\ell_{ij}
∑
i
=
1
n
a
i
≤
∑
1
≤
i
<
j
≤
n
ℓ
ij
?
2
1
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Three polygonal discs
Consider the closed polygonal discs
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
with the property that for any three points
A
∈
P
1
A\in P_1
A
∈
P
1
,
B
∈
P
2
B\in P_2
B
∈
P
2
,
C
∈
P
3
C\in P_3
C
∈
P
3
, we have
[
△
A
B
C
]
≤
1
[\triangle ABC]\le 1
[
△
A
BC
]
≤
1
. (Here
[
X
]
[X]
[
X
]
denotes the area of polygon
X
X
X
.) (a) Prove that
min
{
[
P
1
]
,
[
P
2
]
,
[
P
3
]
}
<
4
\min\{[P_1],[P_2],[P_3]\}<4
min
{[
P
1
]
,
[
P
2
]
,
[
P
3
]}
<
4
. (b) Give an example of polygons
P
1
,
P
2
,
P
3
P_1,P_2,P_3
P
1
,
P
2
,
P
3
with the above property such that
[
P
1
]
>
4
[P_1]>4
[
P
1
]
>
4
and
[
P
2
]
>
4
[P_2]>4
[
P
2
]
>
4
.
1
1
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Similar to Beatty sequences
Let
a
,
b
a,b
a
,
b
be positive real numbers satisfying
2
a
b
=
a
−
b
2ab=a-b
2
ab
=
a
−
b
. Denote for any positive integer
k
k
k
x
k
x_k
x
k
and
y
k
y_k
y
k
to be the closest integer to
a
k
ak
ak
and
b
k
bk
bk
, respectively (if there are two closest integers, choose the larger one). Prove that any positive integer
n
n
n
appears in the sequence
(
x
k
)
k
≥
1
(x_k)_{k\ge 1}
(
x
k
)
k
≥
1
if and only if it appears at least three times in the sequence
(
y
k
)
k
≥
1
(y_k)_{k\ge 1}
(
y
k
)
k
≥
1
.