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Chile Classification NMO
1992 Chile Classification NMO
1992 Chile Classification NMO
Part of
Chile Classification NMO
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1992 Chile Classification / Qualifying NMO IV
p1. Suppose the number
2
+
10
9
3
3
+
2
−
10
9
3
3
\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3}
3
2
+
9
10
3
+
3
2
−
9
10
3
is integer. Calculate it. p2. A house A is located
300
300
300
m from the bank of a
200
200
200
m wide river.
600
600
600
m above and
500
500
500
m from the opposite bank is another house
B
B
B
. A bridge has been built over the river, that allows you to go from one house to the other covering the minimum distance. What distance is this and in what place of the riverbank is at the bridge? p3. Show that it is not possible to find
1992
1992
1992
positive integers
x
1
,
x
2
,
.
.
.
,
x
1992
x_1, x_2, ..., x_{1992}
x
1
,
x
2
,
...
,
x
1992
that satisfy the equation:
∑
j
=
1
1992
2
j
−
1
x
j
1992
=
2
1992
∏
j
=
2
1992
x
j
\sum_{j=1}^{1992}2^{j-1}x_j^{1992}=2^{1992}\prod_{j=2}^{1992}x_j
j
=
1
∑
1992
2
j
−
1
x
j
1992
=
2
1992
j
=
2
∏
1992
x
j
p4. The following division of positive integers with remainder has been carried out. Every cross represents one digit, and each question mark indicates that there are digits, but without specifying how many. Fill in each cross and question mark with the missing digits. https://cdn.artofproblemsolving.com/attachments/8/e/5658e77025210a46723b6d85abf5bdc3c7ad51.png p5. Three externally tangent circles have, respectively, radii
1
1
1
,
2
2
2
,
3
3
3
. Determine the radius of the circumference that passes through the three points of tangency. p6.
∙
\bullet
∙
It is possible to draw a continuous curve that cuts each of the segments of the figure exactly one time? It is understood that cuts should not occur at any of the twelve vertices.
∙
\bullet
∙
What happens if the figure is on a sphere? https://cdn.artofproblemsolving.com/attachments/a/d/b998f38d4bbe4121a41f1223402c394213944f.png p7. Given any
△
A
B
C
\vartriangle ABC
△
A
BC
, and any point
P
P
P
, let
X
1
X_1
X
1
,
X
2
X_2
X
2
,
X
3
X_3
X
3
be the midpoints of the sides
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
respectively, and
Y
1
Y_1
Y
1
,
Y
2
Y_2
Y
2
,
Y
3
Y_3
Y
3
midpoints of
P
C
PC
PC
,
P
A
PA
P
A
,
P
B
PB
PB
respectively. Prove that the segments
X
1
Y
1
X_1Y_1
X
1
Y
1
,
X
2
Y
2
X_2Y_2
X
2
Y
2
,
X
3
Y
3
X_3Y_3
X
3
Y
3
concur.