MathDB

p1. The figure shows four circles of radius

1

, interior and tangent to the larger circle. What is the sum of the areas of shaded regions

1

and

2

? https://cdn.artofproblemsolving.com/attachments/f/1/7c7031369a1ce2567d01f1e48419eb5c9b670a.png
p2.Ana has decided to go out for a walk of exactly one kilometer each day. She lives in a city

5

\times 5

km grid, in which each block measures

100

m and her house is on a center corner. For how many days can you do different tours, if she always begins the tours by leaving her house and ending there as well, but without passing twice for the same point in the route of each day ?.
Note: Day tours different can have parts in common and even determine the same path but in wrong way.
p3. Determine the smallest integer

n

n \ge 4

, for which we can ensure that from n any different integers it is possible to select four of them

a, b, c, d

which have the property that the value of the expression

a+b-c-d

is divisible by

20

.
p4. In the figure, the area of the larger circle is

1

^2

. The smaller circle is tangent to the first circle and to the sides of the inscribed angle measuring

60^o

. What is the area of the smaller circle ? https://cdn.artofproblemsolving.com/attachments/b/d/b35c565976b667ed1dd3ce6068244556714fcd.png
p5. Find all the integers that are the sum of the squares of its four smaller positive divisors.

Problem Statement