MathDB

p1. Two integers

m

and

n

are said to be coprime if there are integers

a

and

b

such that

am + bn = 1

. Show that for each integer

p

, the pair of numbers formed by

21p + 4

and

14p + 3

are always coprime.
p2. Two farmers, Person

A

and Person

B

intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of their respective origins. https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png
p3. The system of equations of four variables is given:

\left\{\begin{array}{l} 23x + 47y - 3z = 434 \\ 47x - 23y - 4w = 183 \\ 19z + 17w = 91 \end{array} \right.

where

x, y, z

, and

w

are positive integers. Determine the value of

(13x - 14y)^3 - (15z + 16w)^3

p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph. https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png Initially the vehicle contains

3

liters of fuel. After two hours, in the journey of fuel remains

1

liter. a. If in

1

liter he can cover a distance of

32

km, what is the distance taken as a whole? Explain why you answered like that? b. After two hours of travel, is there any acceleration or deceleration? Explain your answer. c. Determine what the average speed of the vehicle is.
p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below. https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png He made a rule that the bottom four circles would be filled with positive numbers less than

10

that can be taken from the numbers on the date of his birth, i.e.

26 \,\, - \,\, 12 \,\, - \,\,1961

without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath.
a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain.
b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as

8

circles. He no longer cares whether the numbers are repeated or not .
i. In order to get the smallest value in the top circle, how should the numbers be arranged? ii. How many arrays are worth considering to produce the smallest value?

Problem Statement