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National and Regional Contests
Czech Republic Contests
District Round (Round II)
2011 District Round (Round II)
2011 District Round (Round II)
Part of
District Round (Round II)
Subcontests
(4)
4
1
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Changing numbers on a cube so that they're all equal
Let
M
M
M
be a set of six distinct positive integers whose sum is
60
60
60
. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding
1
1
1
to the numbers on those faces. Determine the number of sets
M
M
M
for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
3
1
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Find m & n such that (m+n)^2 divides 4(mn+1)
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers for which
4
(
m
n
+
1
)
4 (mn +1)
4
(
mn
+
1
)
is divisible by
(
m
+
n
)
2
(m + n)^2
(
m
+
n
)
2
.
1
1
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Eight-digit multiples of 4 with & without the digit 1
Among all eight-digit multiples of four, are there more numbers with the digit
1
1
1
or without the digit
1
1
1
in their decimal representation?
2
1
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Hexagon with twice the area of a triangle
Let
A
B
C
ABC
A
BC
denote a triangle with area
S
S
S
. Let
U
U
U
be any point inside the triangle whose vertices are the midpoints of the sides of triangle
A
B
C
ABC
A
BC
. Let
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
denote the reflections of
A
A
A
,
B
B
B
,
C
C
C
, respectively, about the point
U
U
U
. Prove that hexagon
A
C
′
B
A
′
C
B
′
AC'BA'CB'
A
C
′
B
A
′
C
B
′
has area
2
S
2S
2
S
.