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Chile Classification NMO Juniors
2000 Chile Classification NMO Juniors
2000 Chile Classification NMO Juniors
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Chile Classification NMO Juniors
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2000 Chile Classification / Qualifying NMO Juniors XII
p1. One side of a triangle is equal to one third of the sum of the other two sides. Show that the angle opposite the first side is the smallest of the angles of the triangle. p2. A very vain mathematician's apprentice claimed that he could write any integer positive as a product of fractions of the form
2
q
−
1
q
\frac{2q-1}{q}
q
2
q
−
1
with
q
>
0
q> 0
q
>
0
integer.
∙
\bullet
∙
Prove what said the apprentice is wrong.
∙
\bullet
∙
Tell how you have written the number
49
49
49
. p3. Determine the digits that have been omitted in the multiplication:https://cdn.artofproblemsolving.com/attachments/1/b/eb9a15ba0c019b3a8d909eed7f2f84428a4ca5.png p4. What fractions should be removed from the sum
1
2
+
1
3
+
1
4
+
1
6
+
1
8
+
1
10
+
1
12
\frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac{1}{10} + \frac{1}{12}
2
1
+
3
1
+
4
1
+
6
1
+
8
1
+
10
1
+
12
1
so that the sum is
1
1
1
? Give all the possibilities and explain why there are no more. p5. Let
P
P
P
be a point on side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
. The parallel through
P
P
P
to
A
B
AB
A
B
intersects at side
A
C
AC
A
C
at point
Q
Q
Q
, and the parallel through
P
P
P
to
A
C
AC
A
C
intersects
A
B
AB
A
B
at point
R
R
R
. The ratio between the areas of the triangles
R
B
P
RBP
RBP
and
Q
B
C
QBC
QBC
is
k
2
k^2
k
2
. Determine the ratio of the areas of the triangles
A
R
Q
ARQ
A
RQ
and
A
B
C
ABC
A
BC
. p6. In how many ways is it possible to rearrange the word MATEMATICO so that there are no two adjacent equal letters? p7. Set
A
A
A
has
5
5
5
different numbers. If we do the sum of each pair of numbers from
A
A
A
,
10
10
10
results are obtained:
1977
1977
1977
;
1982
1982
1982
,
1983
1983
1983
,
1984
1984
1984
,
1985
1985
1985
,
1990
1990
1990
,
1993
1993
1993
,
1994
1994
1994
,
1999
1999
1999
and
2001
2001
2001
. What are those
5
5
5
numbers?