MathDB

p1. A set of cards contains

100

cards, each of which is written with a number from

1

up to

100

. On each of the two sides of the card the same number is written, side one is red and the other is green. First of all Leny arranges all the cards with red writing face up. Then Leny did the following three steps: I. Turn over all cards whose numbers are divisible by

2

II. Turn over all the cards whose numbers are divisible by

3

III. Turning over all the cards whose numbers are divisible by

5

, but didn't turn over all cards whose numbers are divisible by

5

and

2

. Find the number of Leny cards now numbered in red and face up,
p2. Find the area of three intersecting semicircles as shown in the following image. https://cdn.artofproblemsolving.com/attachments/f/b/470c4d2b84435843975a0664fad5fee4a088d5.png
p3. It is known that

x+\frac{1}{x}=7

. Determine the value of

A

so that

\frac{Ax}{x^4+x^2+1}=\frac56

.
p4. There are

13

different gifts that will all be distributed to Ami, Ima, Mai,and Mia. If Ami gets at least

4

gifts, Ima and Mai respectively got at least

3

gifts, and Mia got at least

2

gifts, how many possible gift arrangements are there?
p5. A natural number is called a quaprimal number if it satisfies all four following conditions: i. Does not contain zeros. ii. The digits compiling the number are different. iii. The first number and the last number are prime numbers or squares of an integer. iv. Each pair of consecutive numbers forms a prime number or square of an integer.
For example, we check the number

971643

. (i)

971643

does not contain zeros. (ii) The digits who compile

971643

are different. (iii) One first number and one last number of

971643

, namely

9

and

3

is a prime number or a square of an integer. (iv) Each pair of consecutive numbers, namely

97, 71, 16, 64

, and

43

form prime number or square of an integer. So

971643

is a quadratic number.
Find the largest

6

-digit quaprimal number. Find the smallest

6

-digit quaprimal number. Which digit is never contained in any arbitrary quaprimal number? Explain.

Problem Statement