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Chile Classification NMO
1997 Chile Classification NMO
1997 Chile Classification NMO
Part of
Chile Classification NMO
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1997 Chile Classification / Qualifying NMO IX
p1. A vote is made between three candidates:
A
A
A
,
B
B
B
and
C
C
C
, with an electorate of
20
20
20
people. Each elector votes in order for the three candidates, each possible combination receiving at least one vote. Of the voters,
11
11
11
preferred
A
A
A
over
B
B
B
and
12
12
12
preferred
C
C
C
over
A
A
A
. Given this, the electoral commission asks
B
B
B
to withdraw to make an election only between
A
A
A
and
C
C
C
. Faced with this request, the candidate
B
B
B
protests and argues that there are
14
14
14
voters who prefer him over
C
C
C
. Given the confusion, the commission Electoral decides to make an election between the first two largest ace. Determine which of the candidates are the ones who got the first highest ace, and how many votes did they get. p2. For each positive integer
n
n
n
, prove that
n
2
+
n
+
1
n^2 + n + 1
n
2
+
n
+
1
is not a perfect square. p3. Find all integer solutions of the equation
x
3
+
2
y
3
=
4
z
3
x^3 + 2y^3 = 4z^3
x
3
+
2
y
3
=
4
z
3
p4.
△
A
B
C
\vartriangle ABC
△
A
BC
has a circumcircle
K
K
K
. The interior bisectors of the triangle intersect again
K
K
K
at
D
D
D
,
E
E
E
,
F
F
F
.
△
D
E
F
\vartriangle DEF
△
D
EF
turns out to be equilateral. Prove that
△
A
B
C
\vartriangle ABC
△
A
BC
is equilateral. p5. Let
n
≥
2
n\ge 2
n
≥
2
be a natural.
0
≤
x
1
≤
x
2
≤
.
.
.
≤
x
n
0\le x_1\le x_2\le ... \le x_n
0
≤
x
1
≤
x
2
≤
...
≤
x
n
are real, which have sum
1
1
1
. If
x
n
≤
2
3
x_n \le \frac23
x
n
≤
3
2
, prove that there exists
k
k
k
, with
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
, such that
1
3
≤
∑
j
=
1
k
x
j
≤
2
3
\frac13 \le \sum_{j = 1}^{k} x_j \le \frac23
3
1
≤
j
=
1
∑
k
x
j
≤
3
2
p6. Let
p
,
q
,
r
p, q, r
p
,
q
,
r
integers, where
r
r
r
is not a perfect square. Suppose that
x
=
p
+
r
3
+
q
−
r
3
x = \sqrt[3]{p +\sqrt{r}} + \sqrt[3]{q -\sqrt{r}}
x
=
3
p
+
r
+
3
q
−
r
is a rational. Prove that
p
=
q
p = q
p
=
q
. p7. Consider a circle
C
C
C
, with center
O
O
O
and radius
r
>
0
r> 0
r
>
0
. Let
Q
Q
Q
be a point in the interior of
C
C
C
. A point
P
∈
C
P \in C
P
∈
C
is drawn as well as the
∠
O
P
Q
\angle OPQ
∠
OPQ
. Where should point
P
P
P
lie such that the
∠
O
P
Q
\angle OPQ
∠
OPQ
has maximum measure?