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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1974 Polish MO Finals
1974 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
5
1
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binomials not successive terms of an arithmetic progression
Prove that for any natural numbers
n
,
r
n,r
n
,
r
with
r
+
3
≤
n
r + 3 \le n
r
+
3
≤
n
the binomial coefficients
(
n
r
)
n \choose r
(
r
n
)
,
(
n
r
+
1
)
n \choose r+1
(
r
+
1
n
)
,
(
n
r
+
2
)
n \choose r+2
(
r
+
2
n
)
,
(
n
r
+
3
)
n \choose r+3
(
r
+
3
n
)
cannot be successive terms of an arithmetic progression.
6
1
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diagonals in convex n-gon
Several diagonals in a convex
n
n
n
-gon are drawn so as to divide the
n
n
n
-gon into triangles and:(i) the number of diagonals drawn at each vertex is even; (ii) no two of the diagonals have a common interior point.Prove that
n
n
n
is divisible by
3
3
3
.
3
1
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x^2 - rx- 1
Let
r
r
r
be a natural number. Prove that the quadratic trinomial
x
2
−
r
x
−
1
x^2 - rx- 1
x
2
−
r
x
−
1
does not divide any nonzero polynomial whose coefficients are integers with absolute values less than
r
r
r
.
2
1
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salmon in a mountain river must overpass two waterfalls
A salmon in a mountain river must overpass two waterfalls. In every minute, the probability of the salmon to overpass the first waterfall is
p
>
0
p > 0
p
>
0
, and the probability to overpass the second waterfall is
q
>
0
q > 0
q
>
0
. These two events are assumed to be independent. Compute the probability that the salmon did not overpass the first waterfall in
n
n
n
minutes, assuming that it did not overpass both waterfalls in that time.
1
1
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plane _|_ CD , in tetrahedron ABCD
In a tetrahedron
A
B
C
D
ABCD
A
BC
D
the edges
A
B
AB
A
B
and
C
D
CD
C
D
are perpendicular and
∠
A
C
B
=
∠
A
D
B
\angle ACB =\angle ADB
∠
A
CB
=
∠
A
D
B
. Prove that the plane through
A
B
AB
A
B
and the midpoint of the edge
C
D
CD
C
D
, is perpendicular to
C
D
CD
C
D
.
4
1
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Inequalities
Prove that, so have
k
k
k
for
∀
a
1
,
a
2
,
.
.
.
,
a
n
\forall a_1,a_2,...,a_n
∀
a
1
,
a
2
,
...
,
a
n
satisfying
∣
∑
i
=
1
k
a
i
−
∑
j
=
k
+
1
n
a
j
∣
≤
max
1
≤
m
≤
n
∣
a
m
∣
|\sum_{i=1}^k a_i -\sum_{j=k+1}^n a_j |\leq \max_{1\leq m\leq n} |a_m|
∣
i
=
1
∑
k
a
i
−
j
=
k
+
1
∑
n
a
j
∣
≤
1
≤
m
≤
n
max
∣
a
m
∣