MathDB
Problems
Contests
National and Regional Contests
Chile Contests
Chile Classification NMO Juniors
2018 Chile Classification NMO Juniors
2018 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
(1)
1
Hide problems
2018 Chile Classification / Qualifying NMO Juniors XXX
p1. A conical bottle perched on its base is filled with water up to a height that is
8
8
8
cm of its vertex. When the bottle is turned over the water level is at
2
2
2
cm from its base. Calculate the height of the bottle. p2. A square with side
8
8
8
cm is divided into
64
64
64
squares of
1
1
1
cm
2
^2
2
.
7
7
7
little squares are colored black and the rest white. Find the maximum area of a rectangle composed only of small white squares independent of the distribution of the little black squares. p3. From a
1000
1000
1000
-page book, a quantity has been ripped of consecutive of leaves. It is known that the sum of the numbers of the torn pages is
2018
2018
2018
. Determine the numbering of the ripped pages. p4. Given a rhombus
A
B
C
D
ABCD
A
BC
D
, a circle with center at the midpoint of side
A
B
AB
A
B
and with diameter
A
B
AB
A
B
is drawn, which intersects side
B
C
BC
BC
at the point
K
K
K
. Similarly, a circle is drawn with its center at the midpoint of side
A
D
AD
A
D
and of diameter
A
D
AD
A
D
that cuts to the side
C
D
CD
C
D
at point
L
L
L
. Suppose that
∠
A
K
L
=
∠
A
B
C
\angle AKL = \angle ABC
∠
A
K
L
=
∠
A
BC
. Determine the angles of the rhombus sides are equal.PS. Juniors p2, p3 were posted as [url=https://artofproblemsolving.com/community/c4h2691340p23361294]Seniors p2,p1 respectively.