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Chile Classification NMO
1990 Chile Classification NMO
1990 Chile Classification NMO
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Chile Classification NMO
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1990 Chile Classification / Qualifying NMO II
p1. Find the sum of the
120
120
120
numbers
12345
12345
12345
,
12354
12354
12354
,
12435
12435
12435
,
.
.
.
...
...
,
54321
54321
54321
, which are obtained by swapping the digits
1
,
2
,
3
,
4
1, 2, 3, 4
1
,
2
,
3
,
4
,
5
5
5
p2. Find all naturals
n
n
n
such that
n
+
81
2
n
−
5
\frac{n + 81}{2n-5}
2
n
−
5
n
+
81
is a natural number. p3. On each point on the number line, that has the form
a
b
\frac{a}{b}
b
a
(where
a
,
b
a, b
a
,
b
are comprime integers) we draw a circle tangent to the line at that point, with radius
1
2
b
2
\frac{1}{2b^2}
2
b
2
1
. All circles are located in the same half plane. Show that these circles are not secant, and that the circles corresponding to
a
c
\frac{a}{c}
c
a
and
b
d
\frac{b}{d}
d
b
are tangent if and only if
a
d
−
b
c
=
±
1
ad- bc =\pm 1
a
d
−
b
c
=
±
1
.https://cdn.artofproblemsolving.com/attachments/6/d/117cd6c918a64cf17338e587c2e200d76bb073.png p4. Find the minimum number of cuts that need to be made to fully divide, into cubes of
1
1
1
cm of edge, a wooden cube of
4
4
4
cm of edge using a saw that allows only horizontal and vertical cuts, and allows reordering the pieces after each cut. p5. The figure shows three circles of radius
1
1
1
, which are tangent in pairs. Find the area of the region among the three circles (gray in the figure). https://cdn.artofproblemsolving.com/attachments/1/0/e041bf5b4e224f19179fcac2f10e3b474acd31.png p6. Find all pairs of rational numbers
x
x
x
and
y
y
y
that solve the equation:
y
2
−
(
x
−
1
)
x
2
=
0
y^2-(x-1) x^2 = 0
y
2
−
(
x
−
1
)
x
2
=
0
p7. A child receives an amount of money. This money can be received in three different ways :
∙
\bullet
∙
1
1
1
dollar in advance and the rest in monthly installments of
39
39
39
dollars.
∙
\bullet
∙
10
10
10
dollars in advance and the rest in monthly installments of
11
11
11
dollars.
∙
\bullet
∙
2
2
2
dollars in advance and the rest in monthly installments of
7
7
7
dollars. Determine the minimum amount to receive.