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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1970 Polish MO Finals
1970 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
3
1
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C(n,k) is divisible by n, prime cirterion
Prove that an integer
n
>
1
n > 1
n
>
1
is a prime number if and only if, for every integer
k
k
k
with
1
≤
k
≤
n
−
1
1\le k \le n-1
1
≤
k
≤
n
−
1
, the binomial coefficient
(
n
k
)
n \choose k
(
k
n
)
is divisible by
n
n
n
.
6
1
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f' (0) <= A, quadratic , | f(x)| <= 1 for 0 <= x <= 1
Find the smallest real number
A
A
A
such that, for every quadratic polynomial
f
(
x
)
f(x)
f
(
x
)
satisfying
∣
f
(
x
)
∣
≤
1
| f(x)| \le 1
∣
f
(
x
)
∣
≤
1
for
0
≤
x
≤
1
0 \le x \le 1
0
≤
x
≤
1
, it holds that
f
′
(
0
)
≤
A
f' (0) \le A
f
′
(
0
)
≤
A
.
5
1
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psrtition set of 12 elements into six 2-element subsets
In how many ways can a set of
12
12
12
elements be partitioned into six two-element subsets?
4
1
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n rectangles with sides // to 2 perp. lines
In the plane are given two mutually perpendicular lines and
n
n
n
rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.
2
1
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compare terms of 3 sequences
Consider three sequences
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{^\infty}
(
a
n
)
n
=
1
∞
,
(
b
n
)
n
=
1
∞
(b_n)_{n=1}^{^\infty}
(
b
n
)
n
=
1
∞
,
(
c
n
)
n
=
1
∞
(c_n)_{n=1}^{^\infty}
(
c
n
)
n
=
1
∞
, each of which has pairwisedistinct terms. Prove that there exist two indices
k
k
k
and
l
l
l
for which
k
<
l
k < l
k
<
l
,
a
k
<
a
l
,
b
k
<
b
l
,
a
n
d
c
k
<
c
l
.
a_k < a_l , b_k < b_l , \,\,\, and \,\,\, c_k < c_l.
a
k
<
a
l
,
b
k
<
b
l
,
an
d
c
k
<
c
l
.
1
1
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points on semicircles, max sum of triangle areas
Diameter
A
B
AB
A
B
divides a circle into two semicircles. Points
P
1
P_1
P
1
,
P
2
P_2
P
2
,
.
.
.
...
...
,
P
n
P_n
P
n
are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles
C
P
1
P
2
CP_1P_2
C
P
1
P
2
,
C
P
2
P
3
CP_2P_3
C
P
2
P
3
,
.
.
.
...
...
,
C
P
n
−
1
P
n
CP_{n-1}P_n
C
P
n
−
1
P
n
?