MathDB

p1. Among the numbers

\frac15

and

\frac14

there are infinitely many fractional numbers. Find

999

decimal numbers between

\frac15

and

\frac14

so that the difference between the next fractional number with the previous fraction constant. (i.e. If

x_1, x_2, x_3, x_4,..., x_{999}

is a fraction that meant, then

x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}

)
p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png It is known that the area of the triangle in Figure

1

1

unit area. Find the total area of the area formed by the black triangles in figure

5

. Also find the total area of the area formed by the black triangles in the

20

th figure.
p3. For each pair of natural numbers

a

and

b

, we define

a*b = ab + a - b

. The natural number

x

is said to be the constituent of the natural number

n

if there is a natural number

y

that satisfies

x*y = n

. For example,

2

is a constituent of

6

because there is a natural number 4 so that

2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6

. Find all constituent of

2005

.
p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth.
p5. Given the equation

x^2 + 3y^2 = n

, where

x

and

y

are integers. If

n < 20

what number is

n

, and which is the respective pair

(x,y)

? Show that it is impossible to solve

x^2 + 3y^2 = 8

in integers.

Problem Statement