MathDB

p1. Let

A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y

integer numbers

\}

. Find all pairs of

(x, zx)\in A

provided that

z

is non-zero integer.
p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only

4

different weights. (For example, if he has weights

1

2

5

and

10

. He can weighing

1

kg,

2

kg,

3

(1 + 2)

44

(5 - 1)

5

kg,

6

kg,

7

kg,

8

kg,

9

(10 - 1)

10

kg,

11

kg,

12

kg,

13

(10 + 1 + 2)

14

(10 + 5 -1)

15

kg,

16

kg,

17

kg and

18

kg). If he wants to be able to weigh all the weight from

1

kg to

40

kg, determine the four weights that he must have. Explain that your answer is correct.
p3. Given the following table. https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png Table

4\times 4

is a combination of four smaller table sections of size

2\times 2

. This table will be filled with four consecutive integers such that:

\bullet

The horizontal sum of the numbers in each row is

10

\bullet

The vertical sum of the numbers in each column is

10

\bullet

The sum of the four numbers in each part of

2\times 2

which is delimited by the line thickness is also equal to

10

. Determine how many arrangements are possible.
p4. A sequence of real numbers is defined as following:

U_n=ar^{n-1}

, if

n = 4m -3

n = 4m - 2

U_n=- ar^{n-1}

, if

n = 4m - 1

n = 4m

, where

a > 0

r > 0

, and

m

is a positive integer. Prove that the sum of all the

1

st to

2009

th terms is

\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}

5. Cube

ABCD.EFGH

is cut into four parts by two planes. The first plane is parallel to side

ABCD

and passes through the midpoint of edge

BF

. The sceond plane passes through the midpoints

AB

AD

GH

, and

FG

. Determine the ratio of the volumes of the smallest part to the largest part.

Problem Statement