MathDB

p1. Four kite-shaped shapes as shown below (

a > b

a

and

b

are natural numbers less than

10

) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of

16

units of length. What is the possible perimeter of the outermost square formed if it is also known that

a

and

b

are numbers coprime? https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png
p2. If

a = 3^p

b = 3^q

c = 3^r

, and

d = 3^s

and if

p, q, r

, and

s

are natural numbers, what is the smallest value of

p\cdot q\cdot r\cdot s

that satisfies

a^2 + b^3 + c^5 = d^7

3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are

1, 2, 3, 4, 5, 6, 7, 8

, and

9

. ii. The first number used is at least

1

, the second number is at least

2

, third digit-at least

3

, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once.
p 4. For any integer

a, b

, and

c

applies

a\times (b + c) = (a\times b) + (a\times c)

. a) Look for examples that show that

a + (b\times c)\ne (a + b)\times (a + c)

. b) Is it always true that

a + (b\times c) = (a + b)\times(a + c)

? Justify your answer.
p5. The results of a survey of

N

people with the question whether they maintain dogs, birds, or cats at home are as follows:

50

people keep birds,

61

people don't have dogs,

13

people don't keep a cat, and there are at least

74

people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of

N

Problem Statement