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Chile Classification NMO Juniors
2005 Chile Classification NMO Juniors
2005 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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2005 Chile Classification / Qualifying NMO Juniors XVII
p1. Let
a
a
a
,
b
b
b
and
c
c
c
real numbers. If
a
>
b
a> b
a
>
b
and
c
>
d
c> d
c
>
d
, prove that
a
c
+
b
d
>
a
d
+
b
c
ac + bd> ad + bc
a
c
+
b
d
>
a
d
+
b
c
. p2. Prove that the difference of the cubes of two positive integers cannot be
2005
2005
2005
. p3. In the sequence of digits
2005724815826...
2005724815826...
2005724815826...
each term from the fifth is the last digit of the sum of the previous four. Prove that
2006
2006
2006
never appears in this sequence. p4. All the cells on a board of
100
×
100
100 \times 100
100
×
100
are painted in various colors such that no pair of squares with a common side or vertex can have the same color. Determine the minimum number of colors with which it is possible to make such a coloring. p5. Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be a point inside it. Prove that
A
D
+
D
B
<
A
C
+
C
B
AD + DB <AC + CB
A
D
+
D
B
<
A
C
+
CB
. p6. Find the sum of the numbers from
1
1
1
to
2000
2000
2000
that have no
9
9
9
s and no
0
0
0
s in their digits.