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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1989 Romania Team Selection Test
1989 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(5)
4
3
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equipartitionable family of finite sets
A family of finite sets
{
A
1
,
A
2
,
.
.
.
.
.
.
.
,
A
m
}
\left\{ A_{1},A_{2},.......,A_{m}\right\}
{
A
1
,
A
2
,
.......
,
A
m
}
is called equipartitionable if there is a function
φ
:
∪
i
=
1
m
\varphi:\cup_{i=1}^{m}
φ
:
∪
i
=
1
m
→
{
−
1
,
1
}
\rightarrow\left\{ -1,1\right\}
→
{
−
1
,
1
}
such that
∑
x
∈
A
i
φ
(
x
)
=
0
\sum_{x\in A_{i}}\varphi\left(x\right)=0
∑
x
∈
A
i
φ
(
x
)
=
0
for every
i
=
1
,
.
.
.
.
.
,
m
.
i=1,.....,m.
i
=
1
,
.....
,
m
.
Let
f
(
n
)
f\left(n\right)
f
(
n
)
denote the smallest possible number of
n
n
n
-element sets which form a non-equipartitionable family. Prove that a)
f
(
4
k
+
2
)
=
3
f(4k +2) = 3
f
(
4
k
+
2
)
=
3
for each nonnegative integer
k
k
k
, b)
f
(
2
n
)
≤
1
+
m
d
(
n
)
f\left(2n\right)\leq1+m d\left(n\right)
f
(
2
n
)
≤
1
+
m
d
(
n
)
, where
m
d
(
n
)
m d\left(n\right)
m
d
(
n
)
denotes the least positive non-divisor of
n
.
n.
n
.
{A \choose r} denote the family of all r-element subsets of A, function
Let
r
,
n
r,n
r
,
n
be positive integers. For a set
A
A
A
, let
(
A
r
)
{A \choose r}
(
r
A
)
denote the family of all
r
r
r
-element subsets of
A
A
A
. Prove that if
A
A
A
is infinite and
f
:
(
A
r
)
→
1
,
2
,
.
.
.
,
n
f : {A \choose r} \to {1,2,...,n}
f
:
(
r
A
)
→
1
,
2
,
...
,
n
is any function, then there exists an infinite subset
B
B
B
of
A
A
A
such that
f
(
X
)
=
f
(
Y
)
f(X) = f(Y)
f
(
X
)
=
f
(
Y
)
for all
X
,
Y
∈
(
B
r
)
X,Y \in {B \choose r}
X
,
Y
∈
(
r
B
)
.
sphere S remains tangent to a fixed sphere
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be variable points on edges
O
X
,
O
Y
,
O
Z
OX,OY,OZ
OX
,
O
Y
,
OZ
of a trihedral angle
O
X
Y
Z
OXYZ
OX
Y
Z
, respectively. Let
O
A
=
a
,
O
B
=
b
,
O
C
=
c
OA = a, OB = b, OC = c
O
A
=
a
,
OB
=
b
,
OC
=
c
and
R
R
R
be the radius of the circumsphere
S
S
S
of
O
A
B
C
OABC
O
A
BC
. Prove that if points
A
,
B
,
C
A,B,C
A
,
B
,
C
vary so that
a
+
b
+
c
=
R
+
l
a+b+c = R+l
a
+
b
+
c
=
R
+
l
, then the sphere
S
S
S
remains tangent to a fixed sphere.
2
4
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5
1
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The number of latticial cycles
A laticial cycle of length
n
n
n
is a sequence of lattice points
(
x
k
,
y
k
)
(x_k, y_k)
(
x
k
,
y
k
)
,
k
=
0
,
1
,
⋯
,
n
k = 0, 1,\cdots, n
k
=
0
,
1
,
⋯
,
n
, such that
(
x
0
,
y
0
)
=
(
x
n
,
y
n
)
=
(
0
,
0
)
(x_0, y_0) = (x_n, y_n) = (0, 0)
(
x
0
,
y
0
)
=
(
x
n
,
y
n
)
=
(
0
,
0
)
and
∣
x
k
+
1
−
x
k
∣
+
∣
y
k
+
1
−
y
k
∣
=
1
|x_{k+1} -x_{k}|+|y_{k+1} - y_{k}| = 1
∣
x
k
+
1
−
x
k
∣
+
∣
y
k
+
1
−
y
k
∣
=
1
for each
k
k
k
. Prove that for all
n
n
n
, the number of latticial cycles of length
n
n
n
is a perfect square.
3
4
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1
4
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