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Serbia Contests
Serbia National Math Olympiad
2018 Serbia National Math Olympiad
5
5
Part of
2018 Serbia National Math Olympiad
Problems
(1)
Combinatorics on a times b board
Source: Serbia MO 2018 P5
4/2/2018
Let
a
,
b
>
1
a,b>1
a
,
b
>
1
be odd positive integers. A board with
a
a
a
rows and
b
b
b
columns without fields
(
2
,
1
)
,
(
a
−
2
,
b
)
(2,1),(a-2,b)
(
2
,
1
)
,
(
a
−
2
,
b
)
and
(
a
,
b
)
(a,b)
(
a
,
b
)
is tiled with
2
×
2
2\times 2
2
×
2
squares and
2
×
1
2\times 1
2
×
1
dominoes (that can be rotated). Prove that the number of dominoes is at least
3
2
(
a
+
b
)
−
6.
\frac{3}{2}(a+b)-6.
2
3
(
a
+
b
)
−
6.
combinatorics