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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2000 Kurschak Competition
2000 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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A zero-sum problem
Let
k
≥
0
k\ge 0
k
≥
0
be an integer and suppose the integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
give at least
2
k
2k
2
k
different residues upon division by
(
n
+
k
)
(n+k)
(
n
+
k
)
. Show that there are some
a
i
a_i
a
i
whose sum is divisible by
n
+
k
n+k
n
+
k
.
2
1
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Cevians cutting circumcircle in an equilateral triangle
Let
A
B
C
ABC
A
BC
be a non-equilateral triangle in the plane, and let
T
T
T
be a point different from its vertices. Define
A
T
A_T
A
T
,
B
T
B_T
B
T
and
C
T
C_T
C
T
as the points where lines
A
T
AT
A
T
,
B
T
BT
BT
, and
C
T
CT
CT
meet the circumcircle of
A
B
C
ABC
A
BC
. Prove that there are exactly two points
P
P
P
and
Q
Q
Q
in the plane for which the triangles
A
P
B
P
C
P
A_PB_PC_P
A
P
B
P
C
P
and
A
Q
B
Q
C
Q
A_QB_QC_Q
A
Q
B
Q
C
Q
are equilateral. Prove furthermore that line
P
Q
PQ
PQ
contains the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
.
1
1
Hide problems
Coloring lattice points
Paint the grid points of
L
=
{
0
,
1
,
…
,
n
}
2
L=\{0,1,\dots,n\}^2
L
=
{
0
,
1
,
…
,
n
}
2
with red or green in such a way that every unit lattice square in
L
L
L
has exactly two red vertices. How many such colorings are possible?