MathDB
Problems
Contests
National and Regional Contests
Cyprus Contests
Cyprus Team Selection Test
2018 Cyprus IMO TST
4
4
Part of
2018 Cyprus IMO TST
Problems
(1)
Inequality on 2v elements [Cyprus IMO TST 2018]
Source: Cyprus IMO TST 2018, Problem 4
7/1/2018
Let
Λ
=
{
1
,
2
,
…
,
2
v
−
1
,
2
v
}
\Lambda= \{1, 2, \ldots, 2v-1,2v\}
Λ
=
{
1
,
2
,
…
,
2
v
−
1
,
2
v
}
and
P
=
{
α
1
,
α
2
,
…
,
α
2
v
−
1
,
α
2
v
}
P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}
P
=
{
α
1
,
α
2
,
…
,
α
2
v
−
1
,
α
2
v
}
be a permutation of the elements of
Λ
\Lambda
Λ
.(a) Prove that
∑
i
=
1
v
α
2
i
−
1
α
2
i
≤
∑
i
=
1
v
(
2
i
−
1
)
2
i
.
\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.
i
=
1
∑
v
α
2
i
−
1
α
2
i
≤
i
=
1
∑
v
(
2
i
−
1
)
2
i
.
(b) Determine the largest positive integer
m
m
m
such that we can partition the
m
×
m
m\times m
m
×
m
square into
7
7
7
rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14.
1,2,3,4,5,6,7,8,9,10,11,12,13,14.
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14.
inequalities