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Greece Contests
Greece Junior Math Olympiad
1989 Greece Junior Math Olympiad
1989 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
2
1
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no of paths from A to B - Greece Juniors 1989 p3
How many paths are there from
A
A
A
to
B
B
B
that consist of
5
5
5
horizontal segments and
5
5
5
vertical segments of length
1
1
1
each? (see figure) https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png
4
1
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alg. simplifications - Greece Juniors 1989 p4
Simplifyi)
1
+
2
a
+
2
a
a
+
1
a
1+\frac{2a+\dfrac{2}{a}}{a+\dfrac{1}{a}}
1
+
a
+
a
1
2
a
+
a
2
ii)
3
b
+
3
b
+
3
b
2
b
+
1
b
+
1
b
2
\frac{3b+\dfrac{3}{b}+\dfrac{3}{b^2}}{b+\dfrac{1}{b}+\dfrac{1}{b^2}}
b
+
b
1
+
b
2
1
3
b
+
b
3
+
b
2
3
iii)
(
1
a
2
+
1
b
2
+
1
a
b
)
a
6
b
2
−
a
6
−
a
5
b
a
4
b
\frac{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{ab}\right)a^6b^2-a^6-a^5b}{a^4b}
a
4
b
(
a
2
1
+
b
2
1
+
ab
1
)
a
6
b
2
−
a
6
−
a
5
b
3
1
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circle tangent so 2 semicircles, square - Greece Juniors 1989 p3
Given a square
A
B
C
D
ABCD
A
BC
D
of side
a
a
a
, we consider the circle
ω
\omega
ω
, tangent to side
B
C
BC
BC
and to the two semicircles of diameters
A
B
AB
A
B
and
C
D
CD
C
D
. Calculate the radius of circle
ω
\omega
ω
,
1
1
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AB=33333 , sum of 3 consecutives - Greece Juniors 1989 p1
Let
A
A
A
be the sum of three consecutive integers and
B
B
B
be the sum of the exact three consecutive integers. Is it possible to have
A
B
=
33333
AB=33333
A
B
=
33333
?