MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
1997 Turkey Junior National Olympiad
1997 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
Hide problems
Turkey Junior Olympiad 1997, Part II - P3
1
1
1
or
−
1
-1
−
1
is written in
50
50
50
letters. These letters are put into
50
50
50
envelopes. If you ask, you can learn the product of numbers written into any three letters. At least, how many questions are required to find the product of all of the
50
50
50
numbers?
2
1
Hide problems
Turkey Junior Olympiad 1997, Part II - P2
Let
A
B
C
ABC
A
BC
be a triangle with
∣
A
B
∣
=
∣
A
C
∣
=
26
|AB|=|AC|=26
∣
A
B
∣
=
∣
A
C
∣
=
26
,
∣
B
C
∣
=
20
|BC|=20
∣
BC
∣
=
20
. The altitudes of
△
A
B
C
\triangle ABC
△
A
BC
from
A
A
A
and
B
B
B
cut the opposite sides at
D
D
D
and
E
E
E
, respectively. Calculate the radius of the circle passing through
D
D
D
and tangent to
A
C
AC
A
C
at
E
E
E
.
1
1
Hide problems
Turkey Junior Olympiad 1997, Part II - P1
Solve the equation
a
−
a
+
x
=
x
\sqrt {a-\sqrt{a+x}}=x
a
−
a
+
x
=
x
in real numbers in terms of the real number
a
>
1
a>1
a
>
1
.