MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN R Problems
PEN R Problems
Part of
PEN Problems
Subcontests
(12)
12
1
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R 12
Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.
11
1
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R 11
Prove that if a lattice parallelogram contains at most three lattice points in addition to its vertices, then those are on one of the diagonals.
10
1
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R 10
Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.
9
1
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R 9
Prove that if a lattice parallellogram contains an odd number of lattice points, then its centroid.
8
1
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R 8
Prove that on a coordinate plane it is impossible to draw a closed broken line such that [*] coordinates of each vertex are rational, [*] the length of its every edge is equal to
1
1
1
, [*] the line has an odd number of vertices.
7
1
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R 7
Show that the number
r
(
n
)
r(n)
r
(
n
)
of representations of
n
n
n
as a sum of two squares has
π
\pi
π
as arithmetic mean, that is
lim
n
→
∞
1
n
∑
m
=
1
n
r
(
m
)
=
π
.
\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.
n
→
∞
lim
n
1
m
=
1
∑
n
r
(
m
)
=
π
.
6
1
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R 6
Let
R
R
R
be a convex region symmetrical about the origin with area greater than
4
4
4
. Show that
R
R
R
must contain a lattice point different from the origin.
5
1
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R 5
A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is
1
2
\frac{1}{2}
2
1
.
4
1
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R 4
The sidelengths of a polygon with
1994
1994
1994
sides are
a
i
=
i
2
+
4
a_{i}=\sqrt{i^2 +4}
a
i
=
i
2
+
4
(
i
=
1
,
2
,
⋯
,
1994
)
\; (i=1,2,\cdots,1994)
(
i
=
1
,
2
,
⋯
,
1994
)
. Prove that its vertices are not all on lattice points.
3
1
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R 3
Prove no three lattice points in the plane form an equilateral triangle.
2
1
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R 2
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
1
1
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R 1
Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?