2018 Chile Classification / Qualifying NMO Juniors XXX
Source:
October 11, 2021
algebrageometrynumber theorycombinatoricschilean NMO
Problem Statement
p1. A conical bottle perched on its base is filled with water up to a height that is cm of its vertex. When the bottle is turned over the water level is at cm from its base. Calculate the height of the bottle.
p2. A square with side cm is divided into squares of cm. 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 -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 . Determine the numbering of the ripped pages.
p4. Given a rhombus , a circle with center at the midpoint of side and with diameter is drawn, which intersects side at the point . Similarly, a circle is drawn with its center at the midpoint of side and of diameter that cuts to the side at point . Suppose that . 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.