MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1987 Yugoslav Team Selection Test
1987 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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constructing plane intersection
Let there be given lines
a
,
b
,
c
a,b,c
a
,
b
,
c
in the space, no two of which are parallel. Suppose that there exist planes
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
which contain
a
,
b
,
c
a,b,c
a
,
b
,
c
respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)
Problem 2
1
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composition of function involving sqrt(2+sqrt(2))
Let
f
(
x
)
=
2
+
2
x
+
2
−
2
−
2
−
2
x
+
2
+
2
f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}
f
(
x
)
=
−
2
−
2
x
+
2
+
2
2
+
2
x
+
2
−
2
. Find
f
(
f
(
⋯
f
⏟
1987
times
(
x
)
⋯
)
)
\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))
1987
times
f
(
f
(
⋯
f
(
x
)
⋯
))
.
Problem 1
1
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NT recurrence, condition for 7|x_n
Let
x
0
=
a
,
x
1
=
b
x_0=a,x_1=b
x
0
=
a
,
x
1
=
b
and
x
n
+
1
=
2
x
n
−
9
x
n
−
1
x_{n+1}=2x_n-9x_{n-1}
x
n
+
1
=
2
x
n
−
9
x
n
−
1
for each
n
∈
N
n\in\mathbb N
n
∈
N
, where
a
,
b
a,b
a
,
b
are integers. Find the necessary and sufficient condition on
a
a
a
and
b
b
b
for the existence of an
x
n
x_n
x
n
which is a multiple of
7
7
7
.