MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1978 Yugoslav Team Selection Test
1978 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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no element of F is subset of another
Let
F
F
F
be the collection of subsets of a set with
n
n
n
elements such that no element of
F
F
F
is a subset of another of its elements. Prove that
∣
F
∣
≤
(
n
⌊
n
/
2
⌋
)
.
|F|\le\binom n{\lfloor n/2\rfloor}.
∣
F
∣
≤
(
⌊
n
/2
⌋
n
)
.
Problem 2
1
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a sequence of circles, NT?
Let
k
0
k_0
k
0
be a unit semi-circle with diameter
A
B
AB
A
B
. Assume that
k
1
k_1
k
1
is a circle of radius
r
1
=
1
2
r_1=\frac12
r
1
=
2
1
that is tangent to both
k
0
k_0
k
0
and
A
B
AB
A
B
. The circle
k
n
+
1
k_{n+1}
k
n
+
1
of radius
r
n
+
1
r_{n+1}
r
n
+
1
touches
k
n
,
k
0
k_n,k_0
k
n
,
k
0
, and
A
B
AB
A
B
. Prove that:(a) For each
n
∈
{
2
,
3
,
…
}
n\in\{2,3,\ldots\}
n
∈
{
2
,
3
,
…
}
it holds that
1
r
n
+
1
+
1
r
n
−
1
=
6
r
n
−
4
\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4
r
n
+
1
1
+
r
n
−
1
1
=
r
n
6
−
4
. (b)
1
r
n
\frac1{r_n}
r
n
1
is either a square of an even integer, or twice a square of an odd integer.
Problem 1
1
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x^4+x^2y=y^(z+1)
Find all integers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
2
(
x
2
+
y
)
=
y
z
+
1
x^2(x^2+y)=y^{z+1}
x
2
(
x
2
+
y
)
=
y
z
+
1
.