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Problems
Contests
National and Regional Contests
Mexico Contests
Regional Olympiad of Mexico Northwest
2019 Regional Olympiad of Mexico Northwest
2019 Regional Olympiad of Mexico Northwest
Part of
Regional Olympiad of Mexico Northwest
Subcontests
(3)
3
1
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min sum of areas of 2 circles, through, tangent to BC at B and C
On a circle
ω
\omega
ω
with center O and radius
r
r
r
three different points
A
,
B
A, B
A
,
B
and
C
C
C
are chosen. Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be the circles that pass through
A
A
A
and are tangent to line
B
C
BC
BC
at points
B
B
B
and
C
C
C
, respectively. (a) Show that the product of the areas of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
is independent of the choice of the points
A
,
B
A, B
A
,
B
and
C
C
C
. (b) Determine the minimum value that the sum of the areas of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
can take and for what configurations of points
A
,
B
A, B
A
,
B
and
C
C
C
on
ω
\omega
ω
this minimum value is reached.
2
1
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10 friends attend an amusement park, eavh visited 3 different attractions
A group of
10
10
10
friends attend an amusement park. Each has visited three different attractions . Leaving the park and talking to each other, they found that any pair of friends visited at least one attraction in common. Determine what could be the minimum number of friends who could walk in the most visited attraction.
1
1
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2019 different positive integers on the blackboard, winning strategy
Jose and Maria play the following game: Maria writes
2019
2019
2019
positive integers different on the blackboard. Jose deletes some of them (possibly none, but not all) and write to the left of each of the remaining numbers a sign
+
+
+
or a sign
−
-
−
. Then the sum written on the board is calculated. If the result is a multiple of
2019
2019
2019
, Jose wins the game, if not, Maria wins. Determine which of the two has a winning strategy.