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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2020 Kürschák Competition
2020 Kürschák Competition
Part of
Kürschák Math Competition
Subcontests
(3)
P1
1
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discs that CAN be partitioned into at most 10k classes
Let
n
n
n
and
k
k
k
be positive integers. Given
n
n
n
closed discs in the plane such that no matter how we choose
k
+
1
k + 1
k
+
1
of them, there are always two of the chosen discs that have no common point. Prove that the
n
n
n
discs can be partitioned into at most
10
k
10k
10
k
classes such that any two discs in the same class have no common point.
P2
1
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non-negative subadditive FE over the rationals
Find all functions
f
:
Q
→
R
≥
0
f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}
f
:
Q
→
R
≥
0
such that for any two rational numbers
x
x
x
and
y
y
y
the following conditions hold[*]
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
f(x+y)\leq f(x)+f(y)
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
, [*]
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy)=f(x)f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
, [*]
f
(
2
)
=
1
/
2
f(2)=1/2
f
(
2
)
=
1/2
.
P3
1
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santa visits 13 houses in the same order
There are
N
N
N
houses in a city. Every Christmas, Santa visits these
N
N
N
houses in some order. Show that if
N
N
N
is large enough, then it holds that for three consecutive years there are always are
13
13
13
houses such that they have been visited in the same order for two years (out of the three consecutive years). Determine the smallest
N
N
N
for which this holds.