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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1969 Kurschak Competition
1969 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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64 cubes, each with five white faces and one black face
We are given
64
64
64
cubes, each with five white faces and one black face. One cube is placed on each square of a chessboard, with its edges parallel to the sides of the board. We are allowed to rotate a complete row of cubes about the axis of symmetry running through the cubes or to rotate a complete column of cubes about the axis of symmetry running through the cubes. Show that by a sequence of such rotations we can always arrange that each cube has its black face uppermost
2
1
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ABC is equilateral if a(1 - 2 cos A) + b(1 - 2 cos B) + c(1 - 2 cos C) = 0
A triangle has side lengths
a
,
b
,
c
a, b, c
a
,
b
,
c
and angles
A
,
B
,
C
A, B, C
A
,
B
,
C
as usual (with
b
b
b
opposite
B
B
B
etc). Show that if
a
(
1
−
2
cos
A
)
+
b
(
1
−
2
cos
B
)
+
c
(
1
−
2
cos
C
)
=
0
a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0
a
(
1
−
2
cos
A
)
+
b
(
1
−
2
cos
B
)
+
c
(
1
−
2
cos
C
)
=
0
then the triangle is equilateral.
1
1
Hide problems
2 + 2\sqrt{28n^2 + 1} is sqaure if it is an integer
Show that if
2
+
2
28
n
2
+
1
2 + 2\sqrt{28n^2 + 1}
2
+
2
28
n
2
+
1
is an integer, then it is a square (for
n
n
n
an integer).