MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2011 Kurschak Competition
2011 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Distance to lines and points
Given
2
n
2n
2
n
points and
3
n
3n
3
n
lines on the plane. Prove that there is a point
P
P
P
on the plane such that the sum of the distances of
P
P
P
to the
3
n
3n
3
n
lines is less than the sum of the distances of
P
P
P
to the
2
n
2n
2
n
points.
2
1
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Equality between partition numbers
Let
n
n
n
be a positive integer. Denote by
a
(
n
)
a(n)
a
(
n
)
the ways of expression
n
=
x
1
+
x
2
+
…
n=x_1+x_2+\dots
n
=
x
1
+
x
2
+
…
where
x
1
⩽
x
2
⩽
…
x_1\leqslant x_2 \leqslant\dots
x
1
⩽
x
2
⩽
…
are positive integers and
x
i
+
1
x_i+1
x
i
+
1
is a power of
2
2
2
for each
i
i
i
. Denote by
b
(
n
)
b(n)
b
(
n
)
the ways of expression
n
=
y
1
+
y
2
+
…
n=y_1+y_2+\dots
n
=
y
1
+
y
2
+
…
where
y
i
y_i
y
i
is a positive integer and
2
y
i
⩽
y
i
+
1
2y_i\leqslant y_{i+1}
2
y
i
⩽
y
i
+
1
for each
i
i
i
. Prove that
a
(
n
)
=
b
(
n
)
a(n)=b(n)
a
(
n
)
=
b
(
n
)
.
1
1
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Divisibility on a sequence
Let
a
1
,
a
2
,
.
.
.
a_1, a_2,...
a
1
,
a
2
,
...
be an infinite sequence of positive integers such that for any
k
,
ℓ
∈
Z
+
k,\ell\in \mathbb{Z_+}
k
,
ℓ
∈
Z
+
,
a
k
+
ℓ
a_{k+\ell}
a
k
+
ℓ
is divisible by
gcd
(
a
k
,
a
ℓ
)
\gcd(a_k,a_\ell)
g
cd
(
a
k
,
a
ℓ
)
. Prove that for any integers
1
⩽
k
⩽
n
1\leqslant k\leqslant n
1
⩽
k
⩽
n
,
a
n
a
n
−
1
…
a
n
−
k
+
1
a_na_{n-1}\dots a_{n-k+1}
a
n
a
n
−
1
…
a
n
−
k
+
1
is divisible by
a
k
a
k
−
1
…
a
1
a_ka_{k-1}\dots a_1
a
k
a
k
−
1
…
a
1
.