MathDB

p1. A group of mechanics have inspected a lot of old vehicles, and together they determined that

45

of these have engine problems,

30

have electrical faults and

35

have oil leaks. There are exactly

10

vehicles that have engine problems and electrical faults,

10

with engine problems and oil leakage, and

10

with electrical faults and engine leaks. But nevertheless, of all vehicles, only two have all three defects at the same time, while the last

18

vehicles they are in perfect condition. Determine how many total cars are on that lot.
p2. A number is said fourfriend when he and all the numbers that are obtained by rearranging their digits in any order they are multiples of

4

. Determine how many

7

-digit numbers are fourfriends.
Note: A friend number four cannot have

0

digits in its decimal representation.
p3. A sequence of numbers is formed, each of which is equal to the product of

3

consecutive numbers, as follows:

1 \times 2 \times 3

2 \times 3 \times 4

3 \times 4 \times 5

, ... ,

98 \times× 99 \times 10

. Juan adds the inverses of all these numbers and obtains as a result a rational number, whose simplified expression is

p / q

(that is,

p

and

q

have no divisors in common), with natural

p

and

q

. Find the value of

p + q

.
p4. There is a trapezoid

ABCD

with bases

AD

and

BC

. A point

E

moves along side

AB

and let

O_1

O_2

be the centers of the circles that pass through the vertices of the triangles

AED

and

BEC

, respectively. Show that the distance from

O_1

O_2

is constant, regardless of the position of point

E

.
p5. Determine the value of the following expression:

\sqrt{5 + \sqrt{5^2 - 9}} -\sqrt{11 + \sqrt{11^2 - 9}} + \sqrt{17 + \sqrt{17^2 - 9}} - \sqrt{23 + \sqrt{23^2 - 9}} + \sqrt{29 + \sqrt{29^2 - 9}} - ... +\sqrt{797 + \sqrt{797^2-9}}.

Problem Statement