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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
2006 Chile National Olympiad
2006 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
1
1
Hide problems
2 order list of fractions, sum related problem
Juana and Juan have to write each one an ordered list of fractions so that the two lists have the same number of fractions and that the difference between the sum of all the fractions from Juana's list and the sum of all fractions from Juan's list is greater than
123
123
123
. The fractions in Juana's list are
1
2
1
,
2
2
3
,
3
2
5
,
4
2
7
,
5
2
9
,
.
.
.
\frac{1^2}{1}, \frac{2^2}{3},\frac{3^2}{5},\frac{4^2}{7},\frac{5^2}{9},...
1
1
2
,
3
2
2
,
5
3
2
,
7
4
2
,
9
5
2
,
...
And the fractions in John's list are
1
2
3
,
2
2
5
,
3
2
7
,
4
2
9
,
5
2
11
,
.
.
.
\frac{1^2}{3}, \frac{2^2}{5},\frac{3^2}{7},\frac{4^2}{9},\frac{5^2}{11},...
3
1
2
,
5
2
2
,
7
3
2
,
9
4
2
,
11
5
2
,
...
Find the least amount of fractions that each one must write to achieve the objective.
3
1
Hide problems
no of ways to move on a 2x6 grid
We have the following board of
2
×
6
2 \times 6
2
×
6
. [asy] unitsize(0.8 cm);int i;draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2));for (i = 0; i <= 6; ++i) { draw((i,0)--(i,2)); }dot("
A
A
A
", (0,2), NW); dot("
B
B
B
", (6,2), NE); dot("
C
C
C
", (3,0), S); [/asy] Find in how many ways you can go from point
A
A
A
to point
B
B
B
, moving by the segments of the board, respecting the following rules: - You cannot pass through the same point twice. - You can only make three types of movements moving through the segments: To the right, up, down - You have to go through point
C
C
C
.
5
1
Hide problems
row of closed lockers, numbered from 1 to 1024
A bored student walks down a hallway where there is a row of closed lockers, numbered from
1
1
1
to
1024
1024
1024
. Opens cabinet No.
1
1
1
, then skips one cabinet and opens the next, and so on successively. When he reaches the end of the row, he turns around and starts again: he opens the first cabinet it finds closed, he skips the next closed cabinet and so on until the start from the hallway. goes from beginning to end, from end to beginning of the corridor until all the cabinets are left open. What is the number of the last cabinet he opened?
4
1
Hide problems
6-digit number, perfect square and perfect cube,
Let
n
n
n
be a
6
6
6
-digit number, perfect square and perfect cube, if
n
−
6
n -6
n
−
6
is neither even nor multiple of
3
3
3
. Find
n
n
n
.
2
1
Hide problems
integer sides of triangle wanted, give circumradius and area (Chile 2006 L2 p2)
In a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with sides integer numbers, it is known that the radius of the circumcircle circumscribed to
△
A
B
C
\vartriangle ABC
△
A
BC
measures
65
8
\dfrac {65} {8}
8
65
centimeters and the area is
84
84
84
cm². Determine the lengths of the sides of the triangle.
6
1
Hide problems
concurrency from Chile, circumcircles related (2006 L2 p6)
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an acute triangle and scalene, with
B
C
BC
BC
its smallest side. Let
P
,
Q
P, Q
P
,
Q
points on
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively, such that
B
Q
=
C
P
=
B
C
BQ = CP = BC
BQ
=
CP
=
BC
. Let
O
1
,
O
2
O_1, O_2
O
1
,
O
2
be the centers of the circles circumscribed to
△
A
Q
B
,
△
A
P
C
\vartriangle AQB, \vartriangle APC
△
A
QB
,
△
A
PC
, respectively. Sean
H
,
O
H, O
H
,
O
the orthocenter and circumcenter of
△
A
B
C
\vartriangle ABC
△
A
BC
a) Show that
O
1
O
2
=
B
C
O_1O_2 = BC
O
1
O
2
=
BC
. b) Show that
B
O
2
,
C
O
1
BO_2, CO_1
B
O
2
,
C
O
1
and
H
O
HO
H
O
are concurrent