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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1965 Polish MO Finals
1965 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(5)
5
1
Hide problems
triangle inscribed in triangle
Points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
divide respectively the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
in the ratios
k
1
k_1
k
1
,
k
2
k_2
k
2
,
k
3
k_3
k
3
. Calculate the ratio of the areas of triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and
A
B
C
ABC
A
BC
.
4
1
Hide problems
a - b and 2a + 2b + 1 are squares of integers if 2a^2 + a = 3b^2 + b
Prove that if the integers
a
a
a
and
b
b
b
satisfy the equation
2
a
2
+
a
=
3
b
2
+
b
,
2a^2 + a = 3b^2 + b,
2
a
2
+
a
=
3
b
2
+
b
,
then the numbers
a
−
b
a - b
a
−
b
and
2
a
+
2
b
+
1
2a + 2b + 1
2
a
+
2
b
+
1
are squares of integers.
3
1
Hide problems
n points on a circle
n
>
2
n > 2
n
>
2
points are chosen on a circle and each of them is connected to every other by a segment. Is it possible to draw all of these segments in one sequence, i.e. so that the end of the first segment is the beginning of the second, the end of the second - the beginning of the third, etc., and so that the end of the last segment is the beginning of the first?
2
1
Hide problems
x_1^n + x_2^n and x_1^{n+1} + x_2^{n+1} are integer and coprime.
Prove that if the numbers
x
1
x_1
x
1
and
x
2
x_2
x
2
are roots of the equation
x
2
+
p
x
−
1
=
0
x^2 + px - 1 = 0
x
2
+
p
x
−
1
=
0
, where
p
p
p
is an odd number, then for every natural
n
n
n
number
x
1
n
+
x
2
n
x_1^n + x_2^n
x
1
n
+
x
2
n
and
x
1
n
+
1
+
x
2
n
+
1
x_1^{n+1} + x_2^{n+1}
x
1
n
+
1
+
x
2
n
+
1
are integer and coprime.
1
1
Hide problems
\pi /3 <= ( a A+ b B+c C}{a+b+c} < \pi / 2
Prove the theorem: the lengths
a
a
a
,
b
b
b
,
c
c
c
of the sides of a triangle and the arc measures
α
\alpha
α
,
β
\beta
β
,
γ
\gamma
γ
of its opposite angles satisfy the inequalities
π
3
≤
a
α
+
b
β
+
c
γ
a
+
b
+
c
<
π
2
.
\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.
3
π
≤
a
+
b
+
c
a
α
+
b
β
+
c
γ
<
2
π
.