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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
2016 Romania Team Selection Tests
2016 Romania Team Selection Tests
Part of
Romania Team Selection Test
Subcontests
(4)
4
2
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Romania TST 2016 Day 1 P4
Determine the integers
k
≥
2
k\geq 2
k
≥
2
for which the sequence
{
(
2
n
n
)
(
m
o
d
k
)
}
n
∈
Z
≥
0
\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}
{
(
n
2
n
)
(
mod
k
)
}
n
∈
Z
≥
0
is eventually periodic.
Romania TST 2016 Day 2 P4
Given any positive integer
n
n
n
, prove that: (a) Every
n
n
n
points in the closed unit square
[
0
,
1
]
×
[
0
,
1
]
[0,1]\times [0,1]
[
0
,
1
]
×
[
0
,
1
]
can be joined by a path of length less than
2
n
+
4
2\sqrt{n}+4
2
n
+
4
; and (b) There exist
n
n
n
points in the closed unit square
[
0
,
1
]
×
[
0
,
1
]
[0,1]\times [0,1]
[
0
,
1
]
×
[
0
,
1
]
that cannot be joined by a path of length less than
n
−
1
\sqrt{n}-1
n
−
1
.
3
3
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Romania TST 2016 Day 1 P3
Let
n
n
n
be a positive integer, and let
a
1
,
a
2
,
.
.
,
a
n
a_1,a_2,..,a_n
a
1
,
a
2
,
..
,
a
n
be pairwise distinct positive integers. Show that
∑
k
=
1
n
1
[
a
1
,
a
2
,
…
,
a
k
]
<
4
,
\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,
k
=
1
∑
n
[
a
1
,
a
2
,
…
,
a
k
]
1
<
4
,
where
[
a
1
,
a
2
,
…
,
a
k
]
[a_1,a_2,…,a_k]
[
a
1
,
a
2
,
…
,
a
k
]
is the least common multiple of the integers
a
1
,
a
2
,
…
,
a
k
a_1,a_2,…,a_k
a
1
,
a
2
,
…
,
a
k
.
Romania TST 2016 Day 2 P3
Prove that: (a) If
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
is a strictly increasing sequence of positive integers such that
a
2
n
−
1
+
a
2
n
a
n
\frac{a_{2n-1}+a_{2n}}{a_n}
a
n
a
2
n
−
1
+
a
2
n
is a constant as
n
n
n
runs through all positive integers, then this constant is an integer greater than or equal to
4
4
4
; and (b) Given an integer
N
≥
4
N\geq 4
N
≥
4
, there exists a strictly increasing sequene
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
of positive integers such that
a
2
n
−
1
+
a
2
n
a
n
=
N
\frac{a_{2n-1}+a_{2n}}{a_n}=N
a
n
a
2
n
−
1
+
a
2
n
=
N
for all indices
n
n
n
.
Romania TST 2016 Day 3 P3
Given a positive integer
n
n
n
, show that for no set of integers modulo
n
n
n
, whose size exceeds
1
+
n
+
4
1+\sqrt{n+4}
1
+
n
+
4
, is it possible that the pairwise sums of unordered pairs be all distinct.
2
3
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Romania TST 2016 Day 1 P2
Let
n
n
n
be a positive integer, and let
S
1
,
S
2
,
…
,
S
n
S_1,S_2,…,S_n
S
1
,
S
2
,
…
,
S
n
be a collection of finite non-empty sets such that
∑
1
≤
i
<
j
≤
n
∣
S
i
∩
S
j
∣
∣
S
i
∣
∣
S
j
∣
<
1.
\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.
1
≤
i
<
j
≤
n
∑
∣
S
i
∣∣
S
j
∣
∣
S
i
∩
S
j
∣
<
1.
Prove that there exist pairwise distinct elements
x
1
,
x
2
,
…
,
x
n
x_1,x_2,…,x_n
x
1
,
x
2
,
…
,
x
n
such that
x
i
x_i
x
i
is a member of
S
i
S_i
S
i
for each index
i
i
i
.
Romania TST 2016 Day 3 P2
Given a positive integer
k
k
k
and an integer
a
≡
3
(
m
o
d
8
)
a\equiv 3 \pmod{8}
a
≡
3
(
mod
8
)
, show that
a
m
+
a
+
2
a^m+a+2
a
m
+
a
+
2
is divisible by
2
k
2^k
2
k
for some positive integer
m
m
m
.
Function in Z^+
Determine all
f
:
Z
+
→
Z
+
f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+
f
:
Z
+
→
Z
+
such that
f
(
m
)
≥
m
f(m)\geq m
f
(
m
)
≥
m
and
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
f(m+n) \mid f(m)+f(n)
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
for all
m
,
n
∈
Z
+
m,n\in \mathbb{Z}^+
m
,
n
∈
Z
+
1
4
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