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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northeast
2010 Regional Olympiad of Mexico Northeast
2010 Regional Olympiad of Mexico Northeast
Part of
Regional Olympiad of Mexico Northeast
Subcontests
(4)
4
1
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mutual friendships
In a group of people, every two of them have exactly one mutual friend in the group. Prove that there is one person who is friends with all the other people in the group. Note: the friendship is mutual, that is, if
X
X
X
is friends with
Y
Y
Y
, then
Y
Y
Y
is friends with
X
X
X
.
3
1
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IX=IY for I incenter if <BAC=60^o
In triangle
A
B
C
ABC
A
BC
,
∠
B
A
C
=
6
0
o
\angle BAC= 60^o
∠
B
A
C
=
6
0
o
. Angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
meets side
A
C
AC
A
C
at
X
X
X
and angle bisector of
∠
B
C
A
\angle BCA
∠
BC
A
meets side
A
B
AB
A
B
at
Y
Y
Y
. Prove that if
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
, then
I
X
=
I
Y
IX=IY
I
X
=
I
Y
.
2
1
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find x/y with smallest denominator if 41/2010<x/y<1/49
Of all the fractions
x
y
\frac{x}{y}
y
x
that satisfy
41
2010
<
x
y
<
1
49
\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}
2010
41
<
y
x
<
49
1
find the one with the smallest denominator.
1
1
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cut each paper piece into 5 pieces - IX Mexico Regional Northeast 2010 P1
Sofia has
5
5
5
pieces of paper on a table. He takes some of the pieces, cuts each one into
5
5
5
little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with
2010
2010
2010
pieces on the table?