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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1987 Greece Junior Math Olympiad
1987 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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x^{n+1}+ax+b divisible by (x-1)^2 - Greece Juniors 1987 p3
Find real
a
,
b
a,b
a
,
b
such that polynomial
P
(
x
)
=
x
n
+
1
+
a
x
+
b
P(x)=x^{n+1}+ax+b
P
(
x
)
=
x
n
+
1
+
a
x
+
b
to be divisible by
(
x
−
1
)
2
(x-1)^2
(
x
−
1
)
2
. Then find the quotient
P
(
x
)
:
(
x
−
1
)
2
,
n
∈
N
∗
P(x):(x-1)^2 , n\in \mathbb{N}^*
P
(
x
)
:
(
x
−
1
)
2
,
n
∈
N
∗
4
1
Hide problems
x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 - Greece Juniors 1987 p4
If
x
+
y
+
z
=
x
2
+
y
2
+
z
2
=
x
3
+
y
3
+
z
3
=
1
w
i
t
h
x
,
y
,
z
∈
R
,
x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 \ \ with \ \ x,y,z\in \mathbb{R},
x
+
y
+
z
=
x
2
+
y
2
+
z
2
=
x
3
+
y
3
+
z
3
=
1
w
i
t
h
x
,
y
,
z
∈
R
,
prove that at least one of
x
,
y
,
z
x,y,z
x
,
y
,
z
is equal to zero.
2
1
Hide problems
(x-4)(x-5)(x-6)(x-7)=1680 - Greece Juniors 1987 p2
Solve
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
(
x
−
7
)
=
1680
(x-4)(x-5)(x-6)(x-7)=1680
(
x
−
4
)
(
x
−
5
)
(
x
−
6
)
(
x
−
7
)
=
1680
1
1
Hide problems
2 colors for all points of plane - Greece Juniors 1987 p1
We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance
1
1
1
among them.