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Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2015 Chile National Olympiad
2015 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
4
1
Hide problems
no of different numbers in [i^2/2015] (Chile 2015 L2 P4)
Find the number of different numbers of the form
⌊
i
2
2015
⌋
\left\lfloor\frac{i^2}{2015} \right\rfloor
⌊
2015
i
2
⌋
, with
i
=
1
,
2
,
.
.
.
,
2015
i = 1,2, ..., 2015
i
=
1
,
2
,
...
,
2015
.
6
1
Hide problems
2 color painting of closed curve (Chile 2015 L2 P6)
On the plane, a closed curve with simple auto intersections is drawn continuously. In the plane a finite number is determined in this way from disjoint regions. Show that each of these regions can be completely painted either white or blue, so that every two regions that share a curve segment at its edges, they always have different colors.Clarification: a car intersection is simple if looking at a very small disk around from her, the curve looks like a junction
×
\times
×
.
3
1
Hide problems
max no of intersections of circles (Chile 2015 L2 P3)
Consider a horizontal line
L
L
L
with
n
≥
4
n\ge 4
n
≥
4
different points
P
1
,
P
2
,
.
.
.
,
P
n
P_1, P_2, ..., P_n
P
1
,
P
2
,
...
,
P
n
. For each pair of points
P
i
P_i
P
i
,
P
j
P_j
P
j
a circle is drawn such that the segment
P
i
P
j
P_iP_j
P
i
P
j
is a diameter. Determine the maximum number of intersections between circles that can occur, considering only those that occur strictly above
L
L
L
.[hide=original wording]Considere una recta horizontal
L
L
L
con
n
≥
4
n\ge 4
n
≥
4
puntos
P
1
,
P
2
,
.
.
.
,
P
n
P_1, P_2, ..., P_n
P
1
,
P
2
,
...
,
P
n
distintos en ella. Para cada par de puntos
P
i
,
P
j
P_i,P_j
P
i
,
P
j
se traza un circulo de manera tal que el segmento
P
i
P
j
P_iP_j
P
i
P
j
es un diametro. Determine la cantidad maxima de intersecciones entre circulos que pueden ocurrir, considerando solo aquellas que ocurren estrictamente arriba de
L
L
L
.
2
1
Hide problems
primes without multiple ending in 2015 (Chile 2015 L2 P2)
Find all prime numbers that do not have a multiple ending in
2015
2015
2015
.
1
1
Hide problems
construct symmetric wrt point with ungraded ruler (Chile 2015 L2 P1)
On the plane, there is drawn a parallelogram
P
P
P
and a point
X
X
X
outside of
P
P
P
. Using only an ungraded rule, determine the point
W
W
W
that is symmetric to
X
X
X
with respect to the center
O
O
O
of
P
P
P
.
5
1
Hide problems
given cyclic ABCD with metric relations + angle, find radius (Chile 2015 L1 P5)
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. Suppose that
∣
D
A
∣
=
∣
B
C
∣
=
2
|DA| =|BC|= 2
∣
D
A
∣
=
∣
BC
∣
=
2
and
∣
A
B
∣
=
4
|AB| = 4
∣
A
B
∣
=
4
. Let
E
E
E
be the intersection point of lines
B
C
BC
BC
and
D
A
DA
D
A
. Suppose that
∠
A
E
B
=
6
0
o
\angle AEB = 60^o
∠
A
EB
=
6
0
o
and that
∣
C
D
∣
<
∣
A
B
∣
|CD| <|AB|
∣
C
D
∣
<
∣
A
B
∣
. Calculate the radius of the circle.