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Chile Junior Math Olympiad
2009 Chile Junior Math Olympiad
2009 Chile Junior Math Olympiad
Part of
Chile Junior Math Olympiad
Subcontests
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2009 Chile NMO Juniors XXI
[url=https://artofproblemsolving.com/community/c4h1845713p12426358]p1. Consider a triangle whose sides measure
1
,
r
1, r
1
,
r
, and
r
2
r^2
r
2
. Determine all the values of
r
r
r
in such a way that the triangle is right. [url=https://artofproblemsolving.com/community/c4h1845717p12426382]p2. Consider three points inside a square on side
1
1
1
. Show that the area of the triangle they form is less than or equal to
1
2
\frac{1}{2}
2
1
. p3. Find all positive integers
a
,
b
a, b
a
,
b
, that satisfy
a
+
b
=
2009
\sqrt{a} + \sqrt{b} =\sqrt{2009}
a
+
b
=
2009
[url=https://artofproblemsolving.com/community/c4h1845712p12426344]p4. On the base
A
C
AC
A
C
of a triangle angle isosceles
A
B
C
ABC
A
BC
, a point is taken
M
M
M
, so that
∣
A
M
∣
=
p
| AM | = p
∣
A
M
∣
=
p
and
∣
M
C
∣
=
q
| MC | = q
∣
MC
∣
=
q
. The inscribed circles are drawn to the
A
M
B
AMB
A
MB
and
C
M
B
CMB
CMB
triangles, which are tangent to the
B
M
BM
BM
side at points
R
R
R
and
S
S
S
respectively. Find the distance between
R
R
R
and
S
S
S
. [url=https://artofproblemsolving.com/community/c1068820h2935455p26269142]p5. Find a positive integer
x
x
x
, with
x
>
1
x> 1
x
>
1
such that all numbers in the sequence
x
+
1
,
x
x
+
1
,
x
x
x
+
1
,
.
.
.
x + 1,x^x + 1,x^{x^x}+1,...
x
+
1
,
x
x
+
1
,
x
x
x
+
1
,
...
are divisible by
2009.
2009.
2009.
p6. Find the smallest value of
n
n
n
such that
2009
2009
2009
is written as the sum of
n
n
n
perfect cubes.PS. Problem 4 was propopsed also as Seniors P4.