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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1986 Poland - Second Round
1986 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
[A'B'C' ]<= 1/4 [ABC]
In the triangle
A
B
C
ABC
A
BC
, the point
A
′
A'
A
′
on the side
B
C
BC
BC
, the point
B
′
B'
B
′
on the side
A
C
AC
A
C
, the point
C
′
C'
C
′
on the side
A
B
AB
A
B
are chosen so that the straight lines
A
A
′
AA'
A
A
′
,
C
C
′
CC'
C
C
′
intersect at one point, i.e. equivalently
∣
B
A
′
∣
⋅
∣
C
B
′
∣
⋅
∣
A
C
′
∣
=
∣
C
A
′
∣
⋅
∣
A
B
′
∣
⋅
∣
B
C
′
∣
|BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'|
∣
B
A
′
∣
⋅
∣
C
B
′
∣
⋅
∣
A
C
′
∣
=
∣
C
A
′
∣
⋅
∣
A
B
′
∣
⋅
∣
B
C
′
∣
. Prove that the area of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is not greater than
1
/
4
1/4
1/4
of the area of triangle
A
B
C
ABC
A
BC
.
5
1
Hide problems
f(x)f(x + 3) = f(x^2 + x + 3)
Prove that if the polynomial
f
f
f
which is not identical to zero satisfies for every real
x
x
x
the equality
f
(
x
)
f
(
x
+
3
)
=
f
(
x
2
+
x
+
3
)
,
f(x)f(x + 3) = f(x^2 + x + 3),
f
(
x
)
f
(
x
+
3
)
=
f
(
x
2
+
x
+
3
)
,
then it has no real roots .
4
1
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x + y = perfext square if 1/x +1/y=1/z
Natural numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
whose greatest common divisor is equal to 1 satisfy the equation
1
x
+
1
y
=
1
z
\frac{1}{x} + \frac{1}{y} = \frac{1}{z}
x
1
+
y
1
=
z
1
Prove that
x
+
y
x + y
x
+
y
is the square of a natural number.
3
1
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PQ>1, sphere cirucmscribed on a regular tetrahedron
Let S be a sphere cirucmscribed on a regular tetrahedron with an edge length greater than 1. The sphere
S
S
S
is represented as the sum of four sets. Prove that one of these sets includes points
P
P
P
,
Q
Q
Q
such that the length of the segment
P
Q
PQ
PQ
exceeds 1.
2
1
Hide problems
66 players part in chess tournament,
66 players take part in the chess tournament, each player plays one game against each other, and the games take place in four cities. Prove that three players play all their games in the same city.
1
1
Hide problems
2f(2x) = f(x) + x., continuous at 0
Determine all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
continuous at zero and such that for every real number
x
x
x
the equality holds
2
f
(
2
x
)
=
f
(
x
)
+
x
.
2f(2x) = f(x) + x.
2
f
(
2
x
)
=
f
(
x
)
+
x
.