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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2017 Kürschák Competition
2017 Kürschák Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Making Latin square by permuting rows
An
n
n
n
by
n
n
n
table has an integer in each cell, such that no two cells within a row share the same number. Prove that it is possible to permute the elements within each row to obtain a table that has
n
n
n
distinct numbers in each column.
2
1
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p^3-q^2 is linear
Do there exist polynomials
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
with real coefficients such that
p
3
(
x
)
−
q
2
(
x
)
p^3(x)-q^2(x)
p
3
(
x
)
−
q
2
(
x
)
is linear but not constant?
1
1
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Most likely point to be enclosed by uniform cevians
Let
A
B
C
ABC
A
BC
be a triangle. Choose points
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
independently on side segments
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively with a uniform distribution. For a point
Z
Z
Z
in the plane, let
p
(
Z
)
p(Z)
p
(
Z
)
denote the probability that
Z
Z
Z
is contained in the triangle enclosed by lines
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
. For which interior point
Z
Z
Z
in triangle
A
B
C
ABC
A
BC
is
p
(
Z
)
p(Z)
p
(
Z
)
maximised?