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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1990 Greece Junior Math Olympiad
1990 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
1
1
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fibonacci recurrence for juniors - Greece Juniors 1990 p1
Considee thr positive integers
a
1
,
a
2
,
.
.
.
,
a
10
a_1,a_2,...,a_{10}
a
1
,
a
2
,
...
,
a
10
such that from the third and on, each it the sum of it's two previous terms (i.e.
a
3
=
a
2
+
a
1
a_3=a_2+a_1
a
3
=
a
2
+
a
1
,
a
4
=
a
3
+
a
2
a_4=a_3+a_2
a
4
=
a
3
+
a
2
, ...). If
a
5
=
7
a_5=7
a
5
=
7
, find
a
10
a_{10}
a
10
.
2
1
Hide problems
2-var maximum - Greece Juniors 1990 p2
For which real values of
x
,
y
x,y
x
,
y
the expression
2
−
(
x
+
y
3
−
1
)
2
(
x
−
3
2
+
2
y
−
x
3
)
2
+
4
\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}
(
2
x
−
3
+
3
2
y
−
x
)
2
+
4
2
−
(
3
x
+
y
−
1
)
2
becomes maximum? Which is that maximum value?
3
1
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angles and diagonals in reg. 72-gon - Greece Juniors 1990 p3
Let
A
1
A
2
A
3
.
.
.
A
72
A_1A_2A_3...A_{72}
A
1
A
2
A
3
...
A
72
be a regurar
72
72
72
-gon with center
O
O
O
. Calculate an extenral angle of that polygon and the angles
∠
A
45
O
A
46
\angle A_{45} OA_{46}
∠
A
45
O
A
46
,
∠
A
44
A
45
A
46
\angle A_{44} A_{45}A_{46}
∠
A
44
A
45
A
46
. How many diagonals does this polygon have?
4
1
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x^2-\frac{m^2+1}{m -1}x+2m+2=0 - Greece Juniors 1990 p4
For which real values of
m
m
m
does the equation
x
2
−
m
2
+
1
m
−
1
x
+
2
m
+
2
=
0
x^2-\frac{m^2+1}{m -1}x+2m+2=0
x
2
−
m
−
1
m
2
+
1
x
+
2
m
+
2
=
0
has root
x
=
−
1
x=-1
x
=
−
1
?