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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1988 Kurschak Competition
1988 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Show that the triangle contains a nontrivial lattice point
Consider the convex lattice quadrilateral
P
Q
R
S
PQRS
PQRS
whose diagonals intersect at
E
E
E
. Prove that if
∠
P
+
∠
Q
<
18
0
∘
\angle P+\angle Q<180^\circ
∠
P
+
∠
Q
<
18
0
∘
, then the
△
P
Q
E
\triangle PQE
△
PQE
contains inside it or on one of its sides a lattice point other than
P
P
P
and
Q
Q
Q
.
2
1
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A set of triplets from {1,2,...,n}
Set
T
⊂
{
1
,
2
,
…
,
n
}
3
T\subset\{1,2,\dots,n\}^3
T
⊂
{
1
,
2
,
…
,
n
}
3
has the property that for any two triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
and
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
in
T
T
T
, we have
a
<
b
<
c
a<b<c
a
<
b
<
c
, and also, we know that at most one of the equalities
a
=
x
a=x
a
=
x
,
b
=
y
b=y
b
=
y
,
c
=
z
c=z
c
=
z
holds. Maximize
∣
T
∣
|T|
∣
T
∣
.
1
1
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PAB,PBC,PCD,PDA has equal area
Prove that if there exists a point
P
P
P
inside the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that the triangles
P
A
B
PAB
P
A
B
,
P
B
C
PBC
PBC
,
P
C
D
PCD
PC
D
,
P
D
A
PDA
P
D
A
have the same area, then one of the diagonals of
A
B
C
D
ABCD
A
BC
D
bisects the area of the quadrilateral.