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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1989 Kurschak Competition
1989 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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The point cannot return to (1,sqrt2)
We play the following game in a Cartesian coordinate system in the plane. Given the input
(
x
,
y
)
(x,y)
(
x
,
y
)
, in one step, we may move to the point
(
x
,
y
±
2
x
)
(x,y\pm 2x)
(
x
,
y
±
2
x
)
or to the point
(
x
±
2
y
,
y
)
(x\pm 2y,y)
(
x
±
2
y
,
y
)
. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).Prove that starting from the point
(
1
;
2
)
\left(1;\sqrt 2\right)
(
1
;
2
)
, we cannot return to it in finitely many steps.
2
1
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Multiples of M have the same digital sum
For any positive integer
n
n
n
denote
S
(
n
)
S(n)
S
(
n
)
the digital sum of
n
n
n
when represented in the decimal system. Find every positive integer
M
M
M
for which
S
(
M
k
)
=
S
(
M
)
S(Mk)=S(M)
S
(
M
k
)
=
S
(
M
)
holds for all integers
1
≤
k
≤
M
1\le k\le M
1
≤
k
≤
M
.
1
1
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Maximalization + construction problem
In the plane, two intersecting lines
a
a
a
and
b
b
b
are given, along with a circle
ω
\omega
ω
that has no common points with these lines. For any line
ℓ
∣
∣
b
\ell||b
ℓ
∣∣
b
, define
A
=
ℓ
∩
a
A=\ell\cap a
A
=
ℓ
∩
a
, and
{
B
,
C
}
=
ℓ
∩
ω
\{B,C\}=\ell\cap \omega
{
B
,
C
}
=
ℓ
∩
ω
such that
B
B
B
is on segment
A
C
AC
A
C
. Construct the line
ℓ
\ell
ℓ
such that the ratio
∣
B
C
∣
∣
A
B
∣
\frac{|BC|}{|AB|}
∣
A
B
∣
∣
BC
∣
is maximal.