MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1985 Yugoslav Team Selection Test
1985 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 2
1
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angles in parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
E
E
E
be a point in the plane such that
A
E
⊥
A
B
AE\perp AB
A
E
⊥
A
B
and
B
C
⊥
E
C
BC\perp EC
BC
⊥
EC
. Show that either
∠
A
E
D
=
∠
B
E
C
\angle AED=\angle BEC
∠
A
E
D
=
∠
BEC
or
∠
A
E
D
+
∠
B
E
C
=
18
0
∘
\angle AED+\angle BEC=180^\circ
∠
A
E
D
+
∠
BEC
=
18
0
∘
.
Problem 1
1
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sets with conditions, subset of [n]
Suppose each element
i
∈
S
=
{
1
,
2
,
…
,
n
}
i\in S=\{1,2,\ldots,n\}
i
∈
S
=
{
1
,
2
,
…
,
n
}
is assigned a nonempty set
S
i
⊆
S
S_i\subseteq S
S
i
⊆
S
so that the following conditions are fulfilled:(i) for any
i
,
j
∈
S
i,j\in S
i
,
j
∈
S
, if
j
∈
S
i
j\in S_i
j
∈
S
i
then
i
∈
S
j
i\in S_j
i
∈
S
j
; (ii) for any
i
,
j
∈
S
i,j\in S
i
,
j
∈
S
, if
∣
S
i
∣
=
∣
S
j
∣
|S_i|=|S_j|
∣
S
i
∣
=
∣
S
j
∣
then
S
i
∩
S
j
=
∅
S_i\cap S_j=\emptyset
S
i
∩
S
j
=
∅
.Prove that there exists
k
∈
S
k\in S
k
∈
S
for which
∣
S
k
∣
=
1
|S_k|=1
∣
S
k
∣
=
1
.
Problem 3
1
Hide problems
Inequalities
1) proove for positive
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
a
b
+
c
+
b
c
+
d
+
c
d
+
a
+
d
a
+
b
≥
2
\frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} \ge 2
b
+
c
a
+
c
+
d
b
+
d
+
a
c
+
a
+
b
d
≥
2