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Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1986 Greece Junior Math Olympiad
1986 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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<ABD =<DBE=<EBC, <ACD=<DCE=<ECB - Greece Juniors 1986 p3
Inside a triangle ABC, consider points
D
,
E
D, E
D
,
E
such that
∠
A
B
D
=
∠
D
B
E
=
∠
E
B
C
\angle ABD =\angle DBE=\angle EBC
∠
A
B
D
=
∠
D
BE
=
∠
EBC
and
∠
A
C
D
=
∠
D
C
E
=
∠
E
C
B
\angle ACD=\angle DC E=\angle ECB
∠
A
C
D
=
∠
D
CE
=
∠
ECB
. Calculate angles
∠
B
D
E
\angle BDE
∠
B
D
E
,
∠
B
E
C
\angle B EC
∠
BEC
,
∠
D
E
C
\angle D E C
∠
D
EC
in terms of the angle of the triangle
A
B
C
ABC
A
BC
.
4
1
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(x^n+1/x^n) /(x^n-1/x^n) - Greece Juniors 1986 p4
i) If
b
=
a
2
+
1
a
2
a
2
−
1
a
2
b=\dfrac{a^2+ \dfrac{1}{a^2}}{a^2-\dfrac{1}{a^2}}
b
=
a
2
−
a
2
1
a
2
+
a
2
1
, express
c
=
a
4
+
1
a
4
a
4
−
1
a
4
c=\dfrac{a^4+\dfrac{1}{a^4}}{a^4-\dfrac{1}{a^4}}
c
=
a
4
−
a
4
1
a
4
+
a
4
1
, in terms of
b
b
b
.ii) If
k
=
x
n
+
1
x
n
x
n
−
1
x
n
k= \frac{x^{n}+\dfrac{1}{x^{n}}}{x^{n}-\dfrac{1}{x^{n}}}
k
=
x
n
−
x
n
1
x
n
+
x
n
1
, express
m
=
x
2
n
+
1
x
2
n
x
2
n
−
1
x
2
n
m= \frac{x^{2n}+\dfrac{1}{x^{2n}}}{x^{2n}-\dfrac{1}{x^{2n}}}
m
=
x
2
n
−
x
2
n
1
x
2
n
+
x
2
n
1
in terms of
k
k
k
.
2
1
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EZ < largest side of ABC - Greece Juniors 1986 p2
Let
A
B
C
ABC
A
BC
be a triangle. α) If point
D
D
D
lies on side
B
C
BC
BC
, prove that
A
D
<
A
B
AD<AB
A
D
<
A
B
or
A
D
<
A
C
AD <AC
A
D
<
A
C
β) If point
E
E
E
lies on side
A
B
AB
A
B
and point
Z
Z
Z
lies on side
A
C
AC
A
C
, prove that line segment is
E
Z
EZ
EZ
less than largest side of the triangle
A
B
C
ABC
A
BC
.
1
1
Hide problems
(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2 diophantine - Greece Juniors 1986 p1
Find all pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
(
x
+
1
)
(
y
+
1
)
(
x
+
y
)
(
x
2
+
y
2
)
=
16
x
2
y
2
(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2
(
x
+
1
)
(
y
+
1
)
(
x
+
y
)
(
x
2
+
y
2
)
=
16
x
2
y
2