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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1969 Polish MO Finals
1969 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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n points and a circle
Given a set
n
n
n
of points in the plane that are not contained in a single straight line. Prove that there exists a circle passing through at least three of these points, inside which there are none of the remaining points of the set.
4
1
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b >= q + s
Show that if natural numbers
a
,
b
,
p
,
q
,
r
,
s
a,b, p,q,r,s
a
,
b
,
p
,
q
,
r
,
s
satisfy the conditions
q
r
−
p
s
=
1
a
n
d
p
q
<
a
b
<
r
s
,
qr- ps = 1 \,\,\,\,\, and \,\,\,\,\, \frac{p}{q}<\frac{a}{b}<\frac{r}{s},
q
r
−
p
s
=
1
an
d
q
p
<
b
a
<
s
r
,
then
b
≥
q
+
s
.
b \ge q+s.
b
≥
q
+
s
.
2
1
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min sum |x-a_i|
Given distinct real numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
, find the minimum value of the function
y
=
∣
x
−
a
1
∣
+
∣
x
−
a
2
∣
+
.
.
.
+
∣
x
−
a
n
∣
,
x
∈
R
.
y = |x-a_1|+|x-a_2|+...+|x-a_n|, \,\,\, x \in R.
y
=
∣
x
−
a
1
∣
+
∣
x
−
a
2
∣
+
...
+
∣
x
−
a
n
∣
,
x
∈
R
.
1
1
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a/(m+2) + b/(m+1)+ c/m=0
Prove that if real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the equality
a
m
+
2
+
b
m
+
1
+
c
m
=
0
\frac{a}{m+2}+\frac{b}{m+1}+\frac{c}{m}= 0
m
+
2
a
+
m
+
1
b
+
m
c
=
0
for some positive number
m
m
m
, then the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has a root between
0
0
0
and
1
1
1
.
5
1
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a polyhedron having n edges
For which values of n does there exist a polyhedron having
n
n
n
edges?
3
1
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octagon, whose all angles are equal and all sides have rational length
Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.