MathDB

p1. There are

11

large boxes. Some of them contain, each one,

8

medium boxes and the rest are empty. Also, some of the medium boxes contain, each one,

8

small boxes and the rest are empty. All the small boxes are empty. If there are

102

empty boxes, determine the total number of boxes.
p2. If the lengths

a, b

and

c

of the sides of a triangle satisfy the conditions

a + b - c = 2

2ab - c^2 = 4

prove that the triangle is equilateral.
p3. A square of area

A

is divided into

102

smaller squares,

101

of which have sides of length

1

. Determine all possible values for

A

.
p4. In the figure, the ratio of the radius of the circular sector to the radius of the inner circle is

3: 1

. If the circle is tangent to the edges of the sector, determine the ratio of its areas. https://cdn.artofproblemsolving.com/attachments/2/c/6636dee0e9a10da8bde10f06ce297e9faca074.png
p5. A convex polyhedron has

12

squares,

8

hexagons, and

6

octagons for faces. At every vertex exactly one square, one hexagon and one octagon concur. Consider the line segments that join two vertices of the polyhedron and determine how many of them are neither edges nor contained in some face of the polyhedron.

Problem Statement