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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
2015 Romania Team Selection Tests
2015 Romania Team Selection Tests
Part of
Romania Team Selection Test
Subcontests
(5)
5
1
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Maximizing a sum !
Given an integer
N
≥
4
N \geq 4
N
≥
4
, determine the largest value the sum
∑
i
=
1
⌊
k
2
⌋
+
1
(
⌊
n
i
2
⌋
+
1
)
\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)
i
=
1
∑
⌊
2
k
⌋
+
1
(
⌊
2
n
i
⌋
+
1
)
may achieve, where
k
,
n
1
,
…
,
n
k
k, n_1, \ldots, n_k
k
,
n
1
,
…
,
n
k
run through the integers subject to
k
≥
3
k \geq 3
k
≥
3
,
n
1
≥
…
≥
n
k
≥
1
n_1 \geq \ldots\geq n_k\geq 1
n
1
≥
…
≥
n
k
≥
1
and
n
1
+
…
+
n
k
=
N
n_1 + \ldots + n_k = N
n
1
+
…
+
n
k
=
N
.
1
5
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4
3
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Floor function and density !
Let
k
k
k
be a positive integer congruent to
1
1
1
modulo
4
4
4
which is not a perfect square and let
a
=
1
+
k
2
a=\frac{1+\sqrt{k}}{2}
a
=
2
1
+
k
. Show that
{
⌊
a
2
n
⌋
−
⌊
a
⌊
a
n
⌋
⌋
:
n
∈
N
>
0
}
=
{
1
,
2
,
…
,
⌊
a
⌋
}
\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}
{
⌊
a
2
n
⌋
−
⌊
a
⌊
an
⌋
⌋
:
n
∈
N
>
0
}
=
{
1
,
2
,
…
,
⌊
a
⌋
}
.
Pairs of opponents in a parliament !!
Given two integers
h
≥
1
h \geq 1
h
≥
1
and
p
≥
2
p \geq 2
p
≥
2
, determine the minimum number of pairs of opponents an
h
p
hp
h
p
-member parliament may have, if in every partition of the parliament into
h
h
h
houses of
p
p
p
member each, some house contains at least one pair of opponents.
Admissible and null sets !!!
Consider the integral lattice
Z
n
\mathbb{Z}^n
Z
n
,
n
≥
2
n \geq 2
n
≥
2
, in the Euclidean
n
n
n
-space. Define a line in
Z
n
\mathbb{Z}^n
Z
n
to be a set of the form
a
1
×
⋯
×
a
k
−
1
×
Z
×
a
k
+
1
×
⋯
×
a
n
a_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n
a
1
×
⋯
×
a
k
−
1
×
Z
×
a
k
+
1
×
⋯
×
a
n
where
k
k
k
is an integer in the range
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
, and the
a
i
a_i
a
i
are arbitrary integers. A subset
A
A
A
of
Z
n
\mathbb{Z}^n
Z
n
is called admissible if it is non-empty, finite, and every line in
Z
n
\mathbb{Z}^{n}
Z
n
which intersects
A
A
A
contains at least two points from
A
A
A
. A subset
N
N
N
of
Z
n
\mathbb{Z}^n
Z
n
is called null if it is non-empty, and every line in
Z
n
\mathbb{Z}^n
Z
n
intersects
N
N
N
in an even number of points (possibly zero). (a) Prove that every admissible set in
Z
2
\mathbb{Z}^2
Z
2
contains a null set. (b) Exhibit an admissible set in
Z
3
\mathbb{Z}^3
Z
3
no subset of which is a null set .
2
5
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3
5
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