MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
2017 Romania Team Selection Test
2017 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(4)
P3
1
Hide problems
Maximum value of cyclic sum
Given an interger
n
≥
2
n\geq 2
n
≥
2
, determine the maximum value the sum
a
1
a
2
+
a
2
a
3
+
.
.
.
+
a
n
−
1
a
n
\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}
a
2
a
1
+
a
3
a
2
+
...
+
a
n
a
n
−
1
may achieve, and the points at which the maximum is achieved, as
a
1
,
a
2
,
.
.
.
a
n
a_1,a_2,...a_n
a
1
,
a
2
,
...
a
n
run over all positive real numers subject to
a
k
≥
a
1
+
a
2
.
.
.
+
a
k
−
1
a_k\geq a_1+a_2...+a_{k-1}
a
k
≥
a
1
+
a
2
...
+
a
k
−
1
, for
k
=
2
,
.
.
.
n
k=2,...n
k
=
2
,
...
n
P1
3
Hide problems
5-tuples with special property
a) Determine all 4-tuples
(
x
0
,
x
1
,
x
2
,
x
3
)
(x_0,x_1,x_2,x_3)
(
x
0
,
x
1
,
x
2
,
x
3
)
of pairwise distinct intergers such that each
x
k
x_k
x
k
is coprime to
x
k
+
1
x_{k+1}
x
k
+
1
(indices reduces modulo 4) and the cyclic sum
x
0
x
1
+
x
1
x
2
+
x
2
x
3
+
x
3
x
1
\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}
x
1
x
0
+
x
2
x
1
+
x
3
x
2
+
x
1
x
3
is an interger. b)Show that there are infinitely many 5-tuples
(
x
0
,
x
1
,
x
2
,
x
3
,
x
4
)
(x_0,x_1,x_2,x_3,x_4)
(
x
0
,
x
1
,
x
2
,
x
3
,
x
4
)
of pairwise distinct intergers such that each
x
k
x_k
x
k
is coprime to
x
k
+
1
x_{k+1}
x
k
+
1
(indices reduces modulo 5) and the cyclic sum
x
0
x
1
+
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+
x
4
x
0
\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}
x
1
x
0
+
x
2
x
1
+
x
3
x
2
+
x
4
x
3
+
x
0
x
4
is an interger.
2 lines concurrent on the circumcircle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
, let
G
,
H
G,H
G
,
H
be its centroid and otrhocenter. Let
D
D
D
be the otrhogonal projection of
A
A
A
on the line
B
C
BC
BC
, and let
M
M
M
be the midpoint of the side
B
C
BC
BC
. The circumcircle of
A
B
C
ABC
A
BC
crosses the ray
H
M
HM
H
M
emanating from
M
M
M
at
P
P
P
and the ray
D
G
DG
D
G
emanating from
D
D
D
at
Q
Q
Q
, outside the segment
D
G
DG
D
G
. Show that the lines
D
P
DP
D
P
and
M
Q
MQ
MQ
meet on the circumcircle of
A
B
C
ABC
A
BC
.
Number Theory problem on sequence
Let m be a positive interger, let
p
p
p
be a prime, let
a
1
=
8
p
m
a_1=8p^m
a
1
=
8
p
m
, and let
a
n
=
(
n
+
1
)
a
n
−
1
n
a_n=(n+1)^{\frac{a_{n-1}}{n}}
a
n
=
(
n
+
1
)
n
a
n
−
1
,
n
=
2
,
3...
n=2,3...
n
=
2
,
3...
. Determine the primes
p
p
p
for which the products
a
n
(
1
−
1
a
1
)
(
1
−
1
a
2
)
.
.
.
(
1
−
1
a
n
)
a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})
a
n
(
1
−
a
1
1
)
(
1
−
a
2
1
)
...
(
1
−
a
n
1
)
,
n
=
1
,
2
,
3...
n=1,2,3...
n
=
1
,
2
,
3...
are all integral.
P4
1
Hide problems
Smallest area on lattice points
Determine the smallest radius a circle passing through EXACTLY three lattice points may have.
P2
3
Hide problems
Combinatorics from "The editors"
Consider a finite collection of 3-element sets
A
i
A_i
A
i
, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each
A
i
A_i
A
i
has at least one red element.
Classic Problem
Determine all intergers
n
≥
2
n\geq 2
n
≥
2
such that
a
+
2
a+\sqrt{2}
a
+
2
and
a
n
+
2
a^n+\sqrt{2}
a
n
+
2
are both rational for some real number
a
a
a
depending on
n
n
n
Romania TST 5 2017 P2
Let
n
n
n
be a positive integer, and let
S
n
S_n
S
n
be the set of all permutations of
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
. let
k
k
k
be a non-negative integer, let
a
n
,
k
a_{n,k}
a
n
,
k
be the number of even permutations
σ
\sigma
σ
in
S
n
S_n
S
n
such that
∑
i
=
1
n
∣
σ
(
i
)
−
i
∣
=
2
k
\sum_{i=1}^{n}|\sigma(i)-i|=2k
∑
i
=
1
n
∣
σ
(
i
)
−
i
∣
=
2
k
and
b
n
,
k
b_{n,k}
b
n
,
k
be the number of odd permutations
σ
\sigma
σ
in
S
n
S_n
S
n
such that
∑
i
=
1
n
∣
σ
(
i
)
−
i
∣
=
2
k
\sum_{i=1}^{n}|\sigma(i)-i|=2k
∑
i
=
1
n
∣
σ
(
i
)
−
i
∣
=
2
k
. Evaluate
a
n
,
k
−
b
n
,
k
a_{n,k}-b_{n,k}
a
n
,
k
−
b
n
,
k
.* * *