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Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1972 Yugoslav Team Selection Test
1972 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(4)
Problem 4
1
Hide problems
subsets with nonempty intersection
Determine the largest integer
k
(
n
)
k(n)
k
(
n
)
with the following properties: There exist
k
(
n
)
k(n)
k
(
n
)
different subsets of a given set with
n
n
n
elements such that each two of them have a non-empty intersection.
Problem 3
1
Hide problems
inequality in n^2 variables
Assume that the numbers from the table
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋮
a
n
1
a
n
2
⋯
a
n
n
\begin{matrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{matrix}
a
11
a
21
⋮
a
n
1
a
12
a
22
⋮
a
n
2
⋯
⋯
⋯
a
1
n
a
2
n
⋮
a
nn
satisfy the inequality:
∑
j
=
1
n
∣
a
j
1
x
1
+
a
j
2
x
2
+
…
+
a
j
n
x
n
∣
≤
M
,
\sum_{j=1}^n|a_{j1}x_1+a_{j2}x_2+\ldots+a_{jn}x_n|\le M,
j
=
1
∑
n
∣
a
j
1
x
1
+
a
j
2
x
2
+
…
+
a
jn
x
n
∣
≤
M
,
for each choice
x
j
=
±
1
x_j=\pm1
x
j
=
±
1
. Prove that
∣
a
11
+
a
22
+
…
+
a
n
n
∣
≤
M
.
|a_{11}+a_{22}+\ldots+a_{nn}|\le M.
∣
a
11
+
a
22
+
…
+
a
nn
∣
≤
M
.
Problem 2
1
Hide problems
set with two diameters, do they intersect?
If a convex set of points in the line has at least two diameters, say
A
B
AB
A
B
and
C
D
CD
C
D
, prove that
A
B
AB
A
B
and
C
D
CD
C
D
have a common point.
Problem 1
1
Hide problems
complex equation, counting signs
Given non-zero real numbers
u
,
v
,
w
,
x
,
y
,
z
u,v,w,x,y,z
u
,
v
,
w
,
x
,
y
,
z
, how many different possibilities are there for the signs of these numbers if
(
u
+
i
x
)
(
v
+
i
y
)
(
w
+
i
z
)
=
i
?
(u+ix)(v+iy)(w+iz)=i?
(
u
+
i
x
)
(
v
+
i
y
)
(
w
+
i
z
)
=
i
?