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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1966 Poland - Second Round
1966 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
5
1
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trisections of sides, equilaterals outside triang3l
Each of the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
of the triangle
A
B
C
ABC
A
BC
was divided into three equal parts and on the middle sections of these sides as bases, equilateral triangles were built outside the triangle
A
B
C
ABC
A
BC
, the third vertices of which were marked with the letters
A
′
,
B
′
,
C
′
A', B' , C'
A
′
,
B
′
,
C
′
respectively. In addition, points
A
′
′
,
B
′
′
,
C
′
′
A'', B'', C''
A
′′
,
B
′′
,
C
′′
were determined, symmetrical to
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
respectively with respect to the lines
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. Prove that the triangles
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
and
A
′
′
B
′
′
C
′
′
A''B''C''
A
′′
B
′′
C
′′
are equilateral and have the same center of gravity as the triangle
A
B
C
ABC
A
BC
.
3
1
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2 colors for 6 points
6
6
6
points are selected on the plane, none of which
3
3
3
lie on one straight line, and all pairwise segments connecting these points are plotted. Some of the sections are plotted in red and others in blue. Prove that any three of the given points are the vertices of a triangle with sides of the same color.
1
1
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x+y+z+t=xyzt. diophantine
Solve the equation in natural numbers
x
+
y
+
z
+
t
=
x
y
z
t
.
x+y+z+t=xyzt.
x
+
y
+
z
+
t
=
x
yz
t
.
6
1
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sum of squares of projextions is fixed, equilateral
Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line
p
p
p
of the plane of this triangle does not depend on the position of the line
p
p
p
if and only if it the triangle is equilateral.
4
1
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a+1 is perfect square if a^2+a = 3b^2
Prove that if the natural numbers
a
a
a
and
b
b
b
satisfy the equation
a
2
+
a
=
3
b
2
a^2+a = 3b^2
a
2
+
a
=
3
b
2
, then the number
a
+
1
a+1
a
+
1
is the square of an integer.
2
1
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common irrational root in 2 integer cubics
Prove that if two cubic polynomials with integer coefficients have an irrational root in common, then they have another common irrational root.