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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2001 Kurschak Competition
2001 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Smallest lattice triangles' circumcenter not lattice point
In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.
2
1
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Sequences, pairwise different sums
Let
k
≥
3
k\ge 3
k
≥
3
be an integer. Prove that if
n
>
(
k
3
)
n>\binom k3
n
>
(
3
k
)
, then for any
3
n
3n
3
n
pairwise different real numbers
a
i
,
b
i
,
c
i
a_i,b_i,c_i
a
i
,
b
i
,
c
i
(
1
≤
i
≤
n
1\le i\le n
1
≤
i
≤
n
), among the numbers
a
i
+
b
i
a_i+b_i
a
i
+
b
i
,
a
i
+
c
i
a_i+c_i
a
i
+
c
i
,
b
i
+
c
i
b_i+c_i
b
i
+
c
i
, one can find at least
k
+
1
k+1
k
+
1
pairwise different numbers. Show that this is not always the case when
n
=
(
k
3
)
n=\binom k3
n
=
(
3
k
)
.
1
1
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Convex hull not a triangle
3
n
−
1
3n-1
3
n
−
1
points are given in the plane, no three are collinear. Prove that one can select
2
n
2n
2
n
of them whose convex hull is not a triangle.