MathDB

p1. The pattern

ABCCCDDDDABBCCCDDDDABBCCCDDDD...

repeats to infinity. Which letter ranks in place

2533

?
p2. Prove that if

a > 2

and

b > 3

then

ab + 6 > 3a + 2b

.
p3. Given a rectangle

ABCD

with size

16

\times 25

cm,

EBFG

is kite, and the length of

AE = 5

cm. Determine the length of

EF

. https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png
p4. Consider the following series of statements. It is known that

x = 1

. Since

x = 1

then

x^2 = 1

. So

x^2 = x

. As a result,

x^2 - 1 = x- 1

(x -1) (x + 1) = (x - 1) \cdot 1

Using the rule out, we get

x + 1 = 1

1 + 1 = 1

2 = 1

The question. a. If

2 = 1

, then every natural number must be equal to

1

. Prove it. b. The result of

2 = 1

is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong?
p5. To calculate

\sqrt{(1998)(1996)(1994)(1992)+16}

. someone does it in a simple way as follows:

2000^2-2 \times 5\times 2000 + 5^2 - 5

? Is the way that person can justified? Why?
p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than

25,000

Rp.. Behind every coupon is written one of the following numbers:

9

12

42

57

69

21

, 15,

75

24

and

81

. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV

21''

. If the restaurant owner provides as much as

10

21''

TV pieces, how many should be handed over to the the customer?
p7. Given is the shape of the image below. https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png The centers of circles

B

C

D

, and

E

are placed on the diameter of circle

A

and the diameter of circle

B

is the same as the radius of circle

A

. Circles

C

D

, and

E

are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle

A

. What is the ratio of the circumference of the circle

A

with the sum of the circumferences of circles

B

C

D

, and

E

?
p8. It is known that

a + b + c = 0

. Prove that

a^3 + b^3 + c^3 = 3abc

Problem Statement