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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1976 Polish MO Finals
1976 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
1
1
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sin pi/18 .... sin 9pi/18
Is the number
sin
π
18
sin
3
π
18
sin
5
π
18
sin
7
π
18
sin
9
π
18
\sin \frac{\pi}{18} \sin \frac{3\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18} \sin \frac{9\pi}{18}
sin
18
π
sin
18
3
π
sin
18
5
π
sin
18
7
π
sin
18
9
π
rational?
2
1
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a_{n+1} = a_n +b_n 4 recurrence relations
Four sequences of real numbers
(
a
n
)
,
(
b
n
)
,
(
c
n
)
,
(
d
n
)
(a_n), (b_n), (c_n), (d_n)
(
a
n
)
,
(
b
n
)
,
(
c
n
)
,
(
d
n
)
satisfy for all
n
n
n
,
a
n
+
1
=
a
n
+
b
n
,
b
n
+
1
=
b
n
+
c
n
,
a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n,
a
n
+
1
=
a
n
+
b
n
,
b
n
+
1
=
b
n
+
c
n
,
c
n
+
1
=
c
n
+
d
n
,
d
n
+
1
=
d
n
+
a
n
.
c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n.
c
n
+
1
=
c
n
+
d
n
,
d
n
+
1
=
d
n
+
a
n
.
Prove that if
a
k
+
m
=
a
m
,
b
k
+
m
=
b
m
,
c
k
+
m
=
c
m
,
d
k
+
m
=
d
m
a_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m
a
k
+
m
=
a
m
,
b
k
+
m
=
b
m
,
c
k
+
m
=
c
m
,
d
k
+
m
=
d
m
for some
k
≥
1
,
n
≥
1
k\ge 1,n \ge 1
k
≥
1
,
n
≥
1
, then
a
2
=
b
2
=
c
2
=
d
2
=
0
a_2 = b_2 = c_2 = d_2 = 0
a
2
=
b
2
=
c
2
=
d
2
=
0
.
6
1
Hide problems
f(n) = log_p n if f(kl) = f(k)+ f(l)
An increasing function
f
:
N
→
R
f : N \to R
f
:
N
→
R
satisfies
f
(
k
l
)
=
f
(
k
)
+
f
(
l
)
f
o
r
a
l
l
k
,
l
∈
N
.
f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.
f
(
k
l
)
=
f
(
k
)
+
f
(
l
)
f
or
a
ll
k
,
l
∈
N
.
Show that there is a real number
p
>
1
p > 1
p
>
1
such that
f
(
n
)
=
l
o
g
p
n
f(n) =\ log_pn
f
(
n
)
=
l
o
g
p
n
for all
n
n
n
.
3
1
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products of pairs of opposite edges tetrahedron of are sides of a triangle
Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.
5
1
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trawler is about to fish in territorial waters of a neighboring country
A trawler is about to fish in territorial waters of a neighboring country, for what he has no licence. Whenever he throws the net, the coast-guard may stop him with the probability
1
/
k
1/k
1/
k
, where
k
k
k
is a fixed positive integer. Each throw brings him a fish landing of a fixed weight. However, if the coast-guard stops him, they will confiscate his entire fish landing and demand him to leave the country. The trawler plans to throw the net
n
n
n
times before he returns to territorial waters in his country. Find
n
n
n
for which his expected profit is maximal.
4
1
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perp. diagonals in a quad
The diagonals of some quadrilateral with sides
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are perpendicular. Prove that the diagonals of any other quadrilateral with sides
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
also are perpendicular